Here's the transcript:
JELLICO: How many duty watches does the crew stand?
RIKER: We've a standard three shift rotation.
JELLICO: I'd like to change that to four starting tonight. I'd also like to examine the duty roster and the crew evaluations as soon as possible.
JELLICO: I believe we're scheduled to arrive at the Cardassian border during delta shift. Please inform the delta tactical officer that I want to launch a class five probe just before we drop out of warp.
RIKER: I was actually going to talk to you about delta shift a little later, sir. Right now, gamma shift will be on duty when we arrive and I will tell Lieutenant McDowell about the probe.
JELLICO: Is there a problem with delta shift, Will?
RIKER: There is no delta shift yet, sir. I have spoken to the department heads about changing from three shifts to four, and they assure me it's going to cause us significant personnel problems.
JELLICO: So you have not changed the watch rotation.
RIKER: I was going to explain this to you after the ceremony, sir.
JELLICO: You will tell the department heads that as of now the Enterprise is on a four shift rotation. I don't want to talk about it. Get it done. Now that means delta shift will be due to come on duty in two hours. I expect you to have it fully manned and ready when it does. Is that clear?
RIKER: Yes, sir. If you'll excuse me, sir. Captain.
Okay, I assure you that a "standard three shift rotation" does not mean there are only three teams. If it did, nobody would get a day off. Rather, there are at least four but more likely five teams that rotate in a four-on/two-off or five-on/two-ff schedule, working 3x8-hour shifts per day. That would require each team to rotate to a different shift each flip: days this week, nights next week, swings the following week, then back to days. I know because I worked such a schedule most of my 26 years in the Air Force.
Indeed, with a ship the size of the Enterprise-E, it's possible that there are six teams working a three-on/three-off no-flip schedule. But let's say for some stupid reason they gave the ship the minumim number of teams: four. There should be enough manpower to run a 4x6-hour duty schedule. There is no "significant personnel problems", at least not in the short term. Jellico originally ordered the change to happen "starting tonight", which was plenty of lead time to make it happen smoothly. Riker drug his feet and thereby cause it to be more of an issue than it needed to be.
The only valid objection Riker and the department heads should have had was "Okay, sir, but we can't keep this up for very long, or you're going to burn the crew out by not giving them time off." And at that point, Jellico would have looked at him and said, "Of course I know that. I'm not stupid. We're gearing up for a crisis, and once that crisis is over, we can go back to a normal duty schedule."
Been there -- done that. More times than I can count. A no-notice switch to 12-hour shifts for two weeks because someone ended up in the hospital sucks, but you do what you have to to get the job done. Going from a 16x5 to a 24x7 shop to support real-world operations sucks, but the bad guys don't stand down for weekends and hollidays.
P.S> Anyone who'se been in the Navy will tell you that a 3x8-hour schedule is not normal for ship-at-sea duty. The standard is a seven-shift rotation of five four-hour shifts plus two two-hour "dog watch" shifts.