you'll be ok to hang it for a day or so, remember the carcass temp with be approx. 72 F and will take several hours to become ambient be that 40 or 50. The most important thing is get that heat out, which IMO requires skinning, especially in temps of 50 or 60F. if you have warm temps a trip to the freezer or in coolers on ise is better then tainted meat. You do what you have too to protect your investment...JMO.
FYI I put deboned meat from 3 elk in a fridge at elk camp this year, our cooler broke down, without any problems. The trim I sent to the freezer immediately who cares its getting ground at least twice. The straps/loins, tenderloin and hams (sirloins and rounds) were all bagged in trim bags and set in the fridge to cool additionally after become ambient and then cut the following day. The meat is absolutely wonderful. Point is a deer will not take up much room in a fridge or cooler with ice if it's deboned. You don't have to get really picky at this stage just get it off the bone and cooling.

good luck hope you connect with your xbow