If during the quest ‘Into the Wood’ the Steel Key gate is closed and you no longer have the Steel Key for it, change the game difficulty to easy from within the in-game options, and then proceed through the more challenging enemies located on the South path. This should allow you to then proceed to the other end of the Fairie puzzles, where you will be able to enable the Song of Exploration: Struggler's Lament* to open the back door. Though the Steel gate will still be closed in this event, you should now be able to explore the area beyond as normal.
*If Struggler's Lament is not accessible to you, try changing the game difficulty to an easier setting from within the in game options, or recreate your game and toggle the 'Skippable Puzzles' (available via the Exploration Style' settings page) to 'On'. The song, 'Struggler's Lament' will then be accessible to you via your party's Songs page and can be used on glyph-marked doors that are locked by mechanical or magical means.