if you want to stay out your way from supply depot, learn how to survive well, go to depth 14 - 17 arcade elite difficulty, search the ones with lots of treasure boxes (like arena or treasure vaults, avoid graveyard and compound), skip through monsters if you could and only destroy boxes. That would be the most efficient way to get "free" elite orbs of alchemy.
Danger areas from clockwork tunnels are also great, but it's really hard around those depth (harder than regular tier 3, imo), since you have tier 2 damage penalty but tier 3 monsters come out.
Repeating what bluencool had already said, if you can manage your spark of life (not losing too soon), the current system is actually easier to grind than pre-big update.
edit:
oh wait 5-2 hall of heroes is not tier 3, it's 3 star. That means you want to go depth 9 - 12 arcade.
You are giving up too early. Back then, elevators costed 10 energy and we had mist tanks that could store 100 energy at most. We were limited to 10 levels per day, so T2 players would do one of the following:
-Do RJP 3 times a day. Vendor token reward for 3.5k cr. (2500*5+3500=16k cr per 36-48 hours.)
-Do RJP w/o boss 5 times a day. (2300*5=11.5k cr per 24-36 hours)
Both methods earn you enough money to pass 5-2 HoH in 4-7 days. And guess what- completing all 5-2 missions reward you 8 (eight) Advanced Orbs of Alchemy. That covers the helm, armor and shield (Considering you should get at least one orb as a drop, that is.).
The real problem is 6-2 HoH, where missions give you only 1/6 of the orbs you need. But elite orbs are significantly more common than advanced orbs. They have a chance to appear between D10-D22. You will probably get enough orbs for at least two 4* items while going through the missions and playing the game.
And the final BAWS- 8-2 HoH! That was one of the reasons I committed suicide with my old account. One single 5* item would cost 86k cr in total. Therefore, Champion rank(9-1) would cost 344k cr, unless you are using the mission reward recipes (And you should not. At least not the Cobalt set). If you continue on with regular RJP farming, passing the final HoH would take 15-30 days. Hard, but far from impossible.
TL;DR: With missions generously covering a lot of crafting costs, the loot increase from elite difficulty, elevators being free, and pet foods as a good crown source, all the HoH missions can be past without spending a single penny.