Another good strategy is to look for a slime infested floor, especially on the upper levels where they are easy to kill. If you're playing on mobile or pc you can use the stardew predictor to find them, otherwise it's also a bit of luck to stumble onto them.
To add to your suggestion, once you find a slime infested floor, you can keep going back to it on the same day*. For example, if you found a slime infested floor on level 37, and hopefully brought staircases, return to level 35 and staircase back to 37. Repeat as often as you want in that session. You'll want to exit via Leave the Mine instead of using the elevator though.
* I haven't tested by leaving the Mine completely, doing other non-Mine stuff and then returning; only leaving the mine, staying in the Mine lobby, waiting a few seconds for the floor to reset, and going back to a floor. Waiting a few seconds in the lobby
seems to make a difference in repopulating the floor, but I don't know if that's 100% true.