@Kris5902 - Bob is the expert on antibiotics but I have a fair bit of experience with wounds and tissue care (admittedly in mammals who are different).I think you should go with a course of Amoxy-Tyl just in case there is something anaerobic going on. You so far have not treated for that possibility. Let me know if you decide to go that way and need dosing information.
Looking at the picture closely, it looks like the swelling is mainly in the foot and less 'up the leg' than it was previously. This is both good and bad. It suggests that the infection is deep and walled in - maybe in tissue, but possibly in bone. That means anaerobic bacteria is more likely (go with Bob's recommendation on antibiotic) but it also means it will be tough for the antibiotic to reach the infection as it may not have much blood flow.
Birds tend to wall off infection and it becomes a sort of solid mass - so it is possible it is mainly already inactive - though the redness suggests it is still going. Does the swollen part of the foot feel 'tense' or hard? Is it hot? Or does it just feel like regular foot?
Anyway, because of what is likely going on I think you need to try drawing out the infection physically in parallel to using the antibiotics. Particularly as chicken pus (unlike mammalian pus) tends to be more solid and will eventually actually need scraping out.
Ref (worth reading on avian abscesses): https://www.michigan.gov/dnr/0,4570,7-350-79136_79608_85016-26357--,00.html
I know you thought soaking was impossible because of the cold - but I would really encourage you to try if you can figure out how. The goal is to soften the skin so that any infection can make its way out and allow you to pull or scrape out any plugs. Making the soaking solution 'hyper osmolar' (which means more concentrated than his tissues also helps pull fluid out and reduce swelling which will make him more comfortable at least temporarily.
The drawing salve does something similar which is soften the skin so infected material bursts through on its own - again you will end up needing to physically pull/scrape the solid stuff out.
I have marked in green where I think you might try the drawing salve.