Advice for handling deer meat

I just shot my first deer today. I’ve been on several hunts in the past and was taught how to properly field dress, skin, quarter, and butcher big game, but we always had a place to hang the deer for a day or so before starting the butchering process.

I don’t have a good setup for hanging deer at my apartment so I opted to skin and quarter it out in the field and cool it down in a big cooler. When I got home I rinsed and dried the meat off (had some leaves and a bit of dirt on it) filled the cooler with a bunch of ice, added the quarters and backstraps, then added some more ice on top and left the cooler on an incline with the plug open to let the water drain out.

I’m hoping to get it into the freezer as soon as I can since I don’t have a ton of room to let the cooler just sit around inside for a few days and temps outside aren’t supposed to drop until later this evening (going down to 2 C/ 40 F) so maybe I could just stick the cooler outside once it cools off.

My question is how long should I leave the meat to cool before I can start butchering it and putting it in the freezer? Does my method sound okay or is there anything I should change?

Thanks for the advice!