On clipped horses, yes. If I’m riding indoors, I might warm up in a cooler or a quarter sheet over the saddle, then remove it for the meat of the work, then replace it for the cool out walk. If it’s fairly cold (-10 or worse), I’ll put the quarter sheet under the saddle.
For riding a clipped horse outside, the quarter sheet is securely attached under the saddle with girth loops, and the tail cord is in place. Wind and high spirits can wreak havoc on a loose chunk of fabric.
When we get to -25C or worse but my mare needs a walk in the indoor, I’ll tack up and then put a full square wool cooler over the whole shebang so that her neck is also covered. I’ve been crazy enough to even have a little canter with her dressed like that.
She likes her clothes though. I wouldn’t necessarily recommend that get up for every horse. lol
Jumping wise, I think you’d be nuts not to use the quarter sheet under the saddle, with the girth through the girth loops, and the tail cord secured.