I think from your question, you already know the time outs are not working, so I would stop them altogether. When a parenting method isn't being effective, it's important to stop redoing it, because your child will naturally just become more resistant to it. Which sounds the case here.
I never used time outs as punishment.
If my kids were hollering/tantrum (when very young) I removed myself from their presence. The shock that I wasn't going to stand for it made them stop almost immediately (I'd just be in next room, keeping eye on them). When there's no audience, acting up loses its appeal very quickly.
If they were tired, strung out, overstimulated, cranky/whiny, not listening, mean to siblings ... off they went for a few minutes to themselves. I did it when I babysat as a teenager - and I do it as a parent. Still do, even with my older ones.
Sometimes little people can't express how they are feeling and it starts coming out as bad behavior - I see it a lot in kids who don't get downtime. So mine would head off to their room to read, play quietly, etc. I made it sound like you're being crabby, take a minute, when you're ready to be nice you can rejoin us.
Often, they'd be asleep 2 minutes later on their bed (especially around the age they gave up naps). Or they chill out, and ask nicely "May I please come down now?". All the while you're keeping an eye out for them of course (you're not locking them away). My kids appreciate it. Kids don't like being disagreeable - they welcome a parent stepping in and taking control.
If your daughter is acting out and crying a lot, and downtime, maybe more sleep (?), healthy snacks in case she's hungry .. don't help, then mention it to your doctor. Especially where she has been taken out of one daycare already. There are therapists who work specifically with children who can be very helpful. And help you develop skills to deal with her specific behaviors. Often in a session or two.
When my kids defy me, I have a MOMMY VOICE that gets their attention. I mean business. I really didn't have to punish at that age (now that they are in middle school .. that's a different story). But at that age, I agree - redirection. Reward when good. It shouldn't be this big a battle - it just means your approach isn't working so switch it up.
Hope that helps :) Good luck