Once the body is moving better on the longe (I may or may not use side reins...today I did not), I go ahead and mount up. I always use a quarter sheet if the temps are below 25 degrees. I may or may not remove it during my ride, but it stays in place most of the time. The other issue the Wheels presents me with, especially during his "cold backed" days is a very inconsistent contact with the bridle. He's very busy with the contact, which, in turn, leaves me with little control over the shoulders. He tends to have rapid bursts of energy which send him roaring down the long sides on occasion...often with a helicoptering of the head. I have found the best remedy for any horse with inconsistent contact is to bridge my reins and go very forward in all 3 gaits. Bridging the reins allows me to be very still with my hands and I am not tempted to try and fix the neck's squirmy behavior. I simply keep my hand steady and firm, but sympathetic at the same time, and ride very energetically forward. If the horse is playful, so be it. If the horse is spooky, best to have your reins bridged anyway since it will encourage the horse to spook straight and it gives you better balance.
Interestingly, Flagg is very similar to Wheels in his behavior with his connection and contact. Curious if this is a Regazzoni thing since they have so many similarities. It will be interesting to see if Tilly exhibits the same tendencies once she's started under saddle in the next few months. I'll keep you posted!