What time you need to let them out depends a lot on how much room they have in the coop. The more room they have in the coop after they wake up, the more casual you can be about letting them out. Some of that too depends on if you feed and especially water inside or outside the coop.
Dogs may be even more of a threat to yours than foxes. Dogs often are.
Most foxes are mainly nocturnal, especially around human habitat. They are less likely to be seen at night so they adapt to hunting mostly at night. As long as they feel “safe” they are equally at home hunting during the day.
I live away from heavy human habitation. Most of my fox problems have come at dawn or dusk, but I have seen them out during the middle of the day too.