The honest answer is: earlier than most people think, and much earlier if your date is a summer weekend or the last day of the month.
Peak season: May through September
Lease turnovers, closings and school calendars concentrate demand in these months. Four to six weeks of notice is a reasonable target, and Fridays and Saturdays fill first.
Month-end is its own peak
The 29th, 30th and 31st are the busiest dates in any month because most leases and many closings land there. If your date is flexible by even two days, mid-month is easier to book and easier to schedule around.
Off-season: November through March
Winter moves are more available on shorter notice — often two to three weeks. Build in weather contingency: salted walkways, floor protection and a slightly longer schedule.
What to have ready when you book
Both addresses, the date, the home size, whether either end has stairs or an elevator, and any oversized items. That is enough for a meaningful estimate rather than a guess.