In your first walkthrough, you won't get many of the desireable special items (A and S rank) since the game deliberately keeps them rare until you move on to world 2 and 3. So, a decent approach is to push your luck by having a girl in your party who has the "Improved Fortune" skill, which will improve the quality of the stuff you get slightly. On top of that, bringing Tomato with her "Special Item Lover" will force chests to give you more of the items you're looking for, at the cost of missing out on weapons/armors/accessories that might be useful to you as well. Lastly, picking maps with lots of chests like Cheetah Mind or Malgrid 8 will naturally boost the odds of getting something useful by simply improving the quantity of items you'll find per run.
Beyond that, there isn't really much you can do other than min/maxing the time you spend per chest and red orb battle.
Worth noting might also be that the Magnum Bonus usually has an item with a higher rank, and the very last chest in any dungeon (the one chest after the boss battle) is effectively buffed to give you something slightly better as well, so you might want to make sure not to blow that one up by being too lazy to switch to a girl that can open it safely.
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While I'm posting:
Has anyone figured out how to unlock Athena 2.0's "Become Human" skill?
Turns out she can't earn exp in world 3, so removing her items to raise her level cap doesn't suffice, since she can't even earn the single SP needed to make use of a Rocky Chunk. Equally, a can of Skill Again just resets her to her default skills, so that's pointless to try as well.