Yay, I'm happy that your betta got an upgrade and that it is working out! Keep checking the ammonia, nitrites and nitrates daily for a week or so, so that you get a feel what your parameters are.
Regarding the peppered cories, according to the seriously fish profile, they seem to be a relatively larger cory species, growing to 2.5"
https://www.seriouslyfish.com/species/corydoras-paleatus/
I'd be concerned about overstocking your 10g by getting 5-6 of them, though it could be possible if you have a heavily planted tank. Smaller corydoras species include pygmy corydoras, but I'm not sure how their compatibility is with a Betta. This all also depends on your planned water change schedule as well as your hardness, etc.
What are you doing for the 5g to keep it cycled? I think it can go 1-2 weeks with no inhabitants, but after that I think you need to start thinking about keeping the bacteria fed. On the other hand, if you are getting the new betta soon then it won't be an issue.