We went to view a horse who was called 'Grey Mare' which I thought rather sad. She was a very nondescript horse.
The best named racehorse was 'Pedlar of Dreams', she was known as Shirley after Shirley Bassey as she sang out very loudly at breakfast time.
So he shouldn't be fresh then. My other ideas would be: do you warm up the same as you would during the day? Is it likely that your own energy is raised, for example if you have rushed to get to ride, my other half's horse became a complete live wire if he rode immediately on returning from work.
Rather than the uninspiring local riding school I would save for trips to somewhere for schoolmaster type lessons, or weekend riding breaks.
Is there a local riding club where you could volunteer at competitions? It might be a way to get a horsey fix and meet different horsey people, you might...
Things I would consider: has he been clipped recently? Saddle cloth changed or washed? Is it cold or damp from previous use?
I would carefully check all the skin under the saddle for any lumps or bumps that could be missed in a winter coat.
If it continues I would have his saddle checked then...
My 20 yo tb has 3 cups of fast fibre, 2 of spillers hi fibre cubes, one of micronised linseed and a measure or forage plus hoof and skin health twice a day with a good teaspoon of salt in one of the feeds. He is not currently in work and lives out 24/7, grass is still plentiful so no hay yet.
I just worked out that the treads on my saddle are about 30 years old, I doubt they have been cleaned properly for the last 20 🥴 I don't think there are any crevices left to trap dirt. If the stirrups are muddy I wash them under a running tap and dry with a soft cloth.