8 week old Beagle puppy pees in her crate

We took her home about a week ago.

For the first few days, crate training was going well. We have two crates of different sizes from when our cat was having behavior issues and we use both. She has a big crate that she prefers (it's big enough for a fully grown Beagle), and a small crate that's the correct size for her right now according to most crate training recommendations. We put her in the small crate at night and on car rides to get her more used to it. She still whines a bit in the small crate before bed but she does not howl and cry like she did the first two nights.

For the first few nights, I'd wake up at 3 am to let her out and this was enough to prevent accidents. For the past 3 days though she's been peeing in her small crate at different points of the day. Last night, she peed twice in the middle of the night.

We removed all towels/blankets from both crates when we noticed it was happening and wiped everything clean with enzyme spray. We don't let her take water 2 hours before bedtime. We bring her out after she eats, after she plays, and after she wakes up every time. It doesn't make sense to me why she's still peeing in her crate– like sometimes she will be let out, pee outside, then 10 minutes afterwards pee again inside the crate??? We praise her when she pees outside and when she pees inside we interrupt her (if possible) and immediately bring her out again.

What can we do differently? This is our first family dog/puppy so we don't have any experience. I suppose it doesn't help that it's been cold out lately and she is clearly uncomfortable whenever we put her out there.