"We're leaving the galaxy, Mister Mitchell!"
And here are the finished renders of the U.S.S. Enterprise bridge as it appeared in the second pilot episode, "Where No Man Has Gone Before"
Note: In WNMHGB, the bridge command platform is rotated 18 degrees, making it not aligned with the viewscreen as it was elsewhere in the series. I believe this was done for framing purposes, and as such I have moved the conn to it's correct orientation in line with the viewscreen.

Now, to point out some differences between the bridge as it appeared in "WNMHGB" from both "The Cage" and the series proper.
1) The most striking difference between "The Cage" and the "WNMHGB" bridge is the color pallette. Color TV was in its infancy in the mid-60s, and between pilots the producers though to take advantage of the popularity of color TV by injecting loads of primary colors into the set to give it a more fantastical feel.
2) There is no carpeting on this set as there was in the series proper, the floor appearing to be a hard surface. It retains the band of the dark blue-black along the perimeter of the bridge and the command pit, although this band does not fill the floor of the turbolift alcove as it does in "The Cage"
3) The faux metallic silver edges of the command pit from and the rail supports from "The Cage" were painted the blue-black that appears elsewhere in set.
4) The grey and green trim colors of the "Cage" set were entirely replaced by the unmistakable macao orange in WNMHGB, which would remain throughout the series run.
5) The large overhead viewscreens above each console were refined from their "Cage" apperance to be of the configuration we're more familiar with from the series. Either one or two large rectangular displays replaced the smaller displays. Many of the overhead displays featured solid colors in WNMHGB. The "Condition Alert" display above the Engineering Sub-Systems Monitor station appears for the first time here. One display of the Engineering station features a print of a painting of the planet Talos IV, which appeared in the briefing room tabletop viewscreen in "The Cage". This print would later go on to be recolored a deep red for use for the Season One overhead displays.
6) The smaller display screens at eye-level of each perimeter station remain unchanged throughout the remainder of the series. These are actually one of the only aspects of the bridge that was not changed throughout the entire run of TOS, including its pilots.
7) The Burke chairs obviously do not have their added leather backs or their 3 planted-on triangular shaped pieces on the rear of the seat, nor are they painted the light blue like they appear in the series proper.
8) Almost every station retains its "gooseneck" viewer from "The Cage" with the exception of the helm and navigator stations. The displays screens of the gooseneck viewers in WNMHGB are greenish in tint, versus their off-white tint in "The Cage". The gooseneck viewers would be completely removed from the set by "The Corbomite Manuever".
10) The bulkheads flanking the viewscreen recieved red-alert indicators in WNMHGB, and in Season One additional control panels were added to these walls as well.
11) Also absent are the colored lights above each perimeter station, and their respectively colored "vents" below each console from the series proper.
12) The bridge railing in "The Cage" only had openings behind the captain and at the front of the viewscreen. In WNMHGB, two additional openings to the left and right of the conn were added.

The captain's chair was placed up on a pedestal in WNMHGB, giving it a more prominent presence. The gooseneck viewer remianed, but the location of it's attachment was slightly altered, and a placard with the alpha-numeric "3C 42" was placed just above. The control panels altered slightly, with some of the small green "lights" being replaced by an orange one and a larger green one. The left hand side panel remained unchanged. Still absent are the flat intercom units and rocker switches. A datacard slot still is not visible in most shots of the episode, even though
towards the end of the episode it appears a datacard is inserted into the arm chair, however
@feek61 speculates that this was simply a card cut in half and glued to the arm. Anyway, I modeled both versions and can make the slot appear on my model with just a click of a button if I want

10) The viewscreen from "The Cage" was slightly resized to be more compressed vertically. The inner bevel of the viewscreen bulkhead was painted a lighter color. Flashing indicator lights appeared above the viewscreen, and a greeble was added below (
which would later appear, in a different configuration, in the corridor set). The viewscreen was replaced by an entirely different viewscreen before the start of the first season, and went on to be a part of the briefing room set and Starbase 11 Courtroom set in Season One, and was then used in the Auxiliary Control Room set in Seasons 2 and 3.

The helm was repainted from "The Cage", this time adopting the macao orange trim color. This trim color would later be refined in Season One. The center red alert indicator light was shorter and more rectangular than it's wedge-shaped series counterpart. The astrogator graphic was the same, but did not have the 3-pronged "course indicator" guide that showed up in Season One. Also missing from the astrogator are rocker switches, flat intercom panels, and the ship's chronometer which were all added when the series began it's first season. The control panels are the same from "The Cage", with rectangular "aircraft-style" buttons instead of the "primsmatic" or "ice-tray" jeweled buttons we saw later on in seasons one and two, respectively. Also missing is the helmsman's pop-up targeting scanner, which appeared in Season 2.

This is the Communications station, which shared a layout similar to most other bridge perimeter stations. The goose neck viewer is present, but notice the lack of rocker switches, flat intercoms, data-card slots and holders. The control panels also follow the "Pilot" scheme, with the aircraft-style buttons instead of primsatic or ice-tray jeweled buttons.

The Science Station got a slight upgrade in WNMHGB, with the additional of the hooded scope. This scope was painted black in Season One, and was upgraded further with a grey paint scheme and additional greebles in Season 2. Also in Season 2, the scope was replicated at the Engineering and Environment stations. Below the scope, rows of black and white buttons were added (probably buttons from an old adding machine). These buttons were removed in Season One, leaving just the glossy black panel.

A close-up of the Engineering station's left-hand control panels, which remained the same from "The Cage". In fact, all control panel surfaces remained the same as they were in the previous pilot, before being upgraded for the first season. Notice the text on the rectangular buttons. A big thanks to
@feek61, who was a great help at speeding up my process of making all the unique control panels around the bridge. He put together some great infographics of the panels upon my request for this project, which I used as reference. You can view those infographics
on his site!

The stations for Engineering, Environmental, and Engineering Sub-Systems Monitor (ESSM, for short). The ESSM console, which was blank in the cage, recieved a couple of rectangular control panels in WNMHGB. In the episode, these control panels catch fire and explode, and are replaced by beige-colored control panels from the facility on Delta Vega (one of those panels went on to be a wall prop in the corridor set)

The stations for Navigation, Defense, and Defense Sub-Systems. There stations were not seen in either "The Cage" or "WNMHGB", but we can surmise what they appeared like given their later appearances and following the aesthetic of the rest of the Cage bridge. The ship schematic over the DSSM is purely conjecture.

The turbolift alcove changed quite a bit from "The Cage". The door was changed from silver to macao orange, the dedication plaque was added, as well as a red-alert indicator light. The panel at the foot of the door remained from "The Cage" but was omitted after this episode. Something I hadn't noticed before: The door jamb was painted black in WNMHGB, but would be repainted grey by season one.

And, once again, instead of coming up with something elaborate for the ceiling (which we never saw on-screen, save for an episode of TAS), I have just gone with something simple.