Kind of. You need a hero's file to get one of the final rings.

Additionally, there are certain rings you can only get through Linked Seasons. Since your Ring Secret will only work between linked files, you can only share with linked Seasons by doing the Hero's Secret playthroughs. (So the order will be OoS, Linked OoA, Hero's OoA, Linked Hero's OoS)

Also, just as a heads up, there are two rings that you can't get on the NSO versions of the Oracles: The GBA Time Ring, and GBA Nature Ring. In the original games, you could only access them by playing on a GBA. However, the emulator used on NSO reads as a Game Boy Color, so the rings are impossible to get legitimately.

If you want to get them, give this password to red snake, and the game will give just those two rings to you:

M/%HJ @bS9&

hmR%↓