I've attempted to turn out with a cooler before and my poor horsey ended up all tangled up. The cooler I was using had hind leg straps and a buckle on the front but apparently it wasn't enough. Now, if I ever have to leave a cooler on my horse and leave for a while I put on a stretchy surcingle-type thing around their barrel and I found it keeps the cooler in place better.
Also, I'm a big fan of using a blow dryer in winter. I don't know your situation but I have access to a barn with electrical outlets we can use so I just blow dry my horse off. Now, some can't stand it and others really enjoy it.
Good luck with your new horse!
"You know, for as long as I can remember, I've had memories." ~Colin Mochrie