In my game, morels showed up a number of times, in the Secret Woods area. I used to hold frequent trips there, for the hardwood, but both mushrooms and fiddlehead ferns, would sometimes show up. I got more than enough of anything to cook, tailor or utilize (community bundle? npc help? etc).
I made the farm cave into a fruit one and that, at least for me, worked wonders! I got a huge variety out of everything, especially the tree products and even out of season, ones! I also got the typical forage-able items (such as the couple of those berries you utilize better post the "bear knowledge" secret), spice berries and other stuff. Fruit cave was what helped me most, towards completing the community bundle (exotic?), too!
Maybe it's strictly RNG, yet every couple of days, the cave has something to pick up. And if i neglect visitting for a weak, then most of the times, the entire floor is littered with various fruit, even! Don't quote me on that, but Fruit Cave is the best, by far!