Pretty much the collar for the bounty. There's a case to be made for just doing everything normally(with maybe a collar use on whiteoak to get Aura through act 1 faster). You ideally want to complete the festival of greed before day 30 though, and the combination of its bounty, giant's bone, area unlock, bonus salary, and exp trickle are a really solid way to speed up your run.
The hp/mp reset combined with fire, NG+ stats, and level 2 should also be enough for Goblin Shaman IIRC(not sure if this relied on RNG or Water Skin though)
Central forest + bandit camp(sneak through the tunnel to get behind the bandit leader) + North forest Scorpion cave + refugee camp... ah right, I suppose the Northern mines would require a day 2.
It's just one day, true. It's still enough of a waste that I generally choose to either pay up or wait since getting those early equipment upgrades is so impactful. Allied npcs(slime doesn't count, slime MVP) on nightmare are largely useless except as meatshields and buff monkeys and you should be able to clean everything up to the minotaur with just John and Paul by the time Charlotte gets out of jail.
If you've switched to avoiding consumables, disregard what I said on those. I thought you were still doing a mostly pure run where the alchemist discount was treated as a necessary sacrifice and consumables were used for certain bosses.