May I ask how old are you? If you are older by my calculations, then you might not be as comfortable with riding stirrup less or bareback, something Sorrelhorse might not think of, since she is young and spry (aren't you SH?).
It is good advice but if that's not in the cards for you, then consider somethig else.
First of all, I think you might be tightening muscles that you shouldn't be , because one should not be tired in the calf muscles by the end of an hour lesson. What are you doing that is making them tired? I mean, you should be riding from your seat primarily, (certainly at the walk and sitting trot, or canter) . If you are posting the trot, it's possible that your calf muscle might get tired, especially if you are not used to riding.
But for the most part, that muscle should not be used all that much. For getting the hrose to move you do not want to be "squeezing" so much as you want to be "plumping". I mean , think of the hrose like a pillow that you plump with your ankles. Don't squeeaze , you kind of flutter your lower leg against the hrose..
If that doesnt' get some action, then you need to carry a whip or take the long tail ends of the reins if there are long ends, and give the hrose a good swat. First ask nicely, then bob with you ankles, then swat with the whip or long rein ends.
If your calves are tired, you are doing too much work down below. Sit down in the saddle on your seatbones, relax , let the weight go down into you heels (don't force it) and plump that pillow of a hrose to get him to go!