Morel can be found in the Cindersap Forest area with Master Canoli's statue. There, you can also find Fiddlehead Ferns; before 1.6, you could get those only there, before reaching the ginger island. You sometimes find the other rare mushroom, Chanterelle.
Hey, if you need morel out of season, you can always plant a small *Oak* trees orchard and put some mushroom logs, right in the middle! To be sure you only get morels and no other mushrooms, just cut all other threes in the near vicinity!
*Pine* trees yield Chanterelle. *Mystic* trees yield purple mushrooms.
*Mapple* trees yield 90% red and 10% purple, mushrooms. *Mahogany* trees, yield common mushrooms.
You need neither cave, nor certain season, anymore. Just carefully planting wild trees and putting mushroom logs strategically, in their midst. I made a mini wild trees orchard in the railroad, complete with stone pavement to prevent seed spreading and tappers, to maximize profit and production!