Usually they stop visiting escorts around a week before, although there will always be some visiting until the last few days before it begins. Although they are allowed to have sex during the night, it is usually with their wives. Some will visit some escort during the night, but very few.
In an area with a Muslim population the decline in business is notable.

What I have experience off is that people will meet up at 10pm ish / sundown to have a communal meal.
It maybe doesn't apply to married with young kids guys who need to do the "kids to bed" routine but for everyone else the sundown meal is very social so single guys might have trouble making excuses to disappear from friends/ parents etc for a late booking even when they are allowed/ or chose to be a bit  more relaxed from the religious rules.
They could be looking for midnight time for a meeting which not many of us seem to work