OK, here's a tip from a seasoned kitten wrangler.Lesson learned… hand tame the kitten first Then let it out. Lol.
Get one of those long wand toys with either a long furry snake thing or some long dangly feathers on it. Start by playing with the kitten through the bars of the cage. You can stay hidden and just get them focused on the toy. After a while (could be a few days or longer, depending on the kitten) they'll start playing even with you in sight. You can then practice leading the kitten out of the crate and around the room with the toy. Once they've got comfortable with that you can always use the toy to lead them back into their room or crate whenever you need to.
The trick is to always make sure the kitten sees playtime as something positive. Stay relaxed about it, even if you're in a hurry to get the kitten back in their crate. Make them see going back into their crate after playtime as a positive thing too. Have some yummy treats to reward them with once they go back in. Something like Churu worked well with my last lot of feral kittens. You should only give that treat at that time. Then they'll learn playtime, then crate time means treats.