Freelance project start dates when you're fully booked
What to tell a new client while your current work is waiting for sign-off.

Promise a start date you can keep even if your current project takes longer than planned. If you can start sooner once a client signs off, offer the earlier date provisionally and agree when you'll confirm it.
Which start date can you commit to?
Use the first date that leaves room for the work you have already agreed to finish, including any revisions you still owe. Check what the new client needs in the first week. You might have time for a call on Monday even if the work that follows needs two days you can't yet set aside. Ask which of those the client means by starting.

When you plan the week around work you've already agreed to deliver, put the hours for those possible revisions on the calendar. If they overlap with the new work, move the proposed start or agree with the new client on a smaller first step that fits.
How do you offer a provisional start date?
Agree on the last time the new client can wait for an answer. Give them a date they can plan around if the earlier slot falls through.
In a hypothetical example, you're an independent consultant waiting for sign-off on a report on Thursday the 5th. The new client wants to start on Monday the 9th, and you may still need that Monday and Tuesday for revisions. You've checked your diary and can set aside Monday the 16th. If the client can wait until Friday to arrange their kickoff, you could say:
“I can start on Monday the 16th. Monday the 9th may open up if my current client signs off on Thursday. I'll confirm by noon Friday the 6th so you know which date to book.”

If they need to book their team by Wednesday, give them the later date on Wednesday. They can then decide whether that still works for their project.
What if this happens with most new clients?
Look at which part of your current work keeps moving. If client review regularly spills into the next booking, leave time for revisions when you plan the next job. If the overlap comes from taking on assignments you meant to turn down, business coaching is a place to work through which clients to say yes to and how much of your week you want to give them.