Since I couldn't find any other way to find the secret exit of Grass and Dirt, I shell-jumped. BLARGH.
Edit: I got the normal exit of the secret level after it, but all it did was make random tiles appear on the overworld, and I can't find any secret exit.
Oh. I've got to make that jump higher, then.
You're supposed to get the Silver P-Switch, take it to the pipe with Munchers over it, get the Blue P-Switch from in there, take it to the area with the Silver P-Switch, use the Silver P-Switch to enter the pipe to the left, bring the Blue P-Switch into the Coin Heaven, wait past the exit pipe, use the Blue P-Switch to get the trampoline, and take the trampoline to the uprise to reach the area with the secret exit.
That's what the Area Map does - if you remember the cutscene at the end of "The Guard Fortress?", it reveals the locations of the levels in that specific area and the directions in which they lead. You can use those tiles to determine which levels have a secret exit.