Easy to find might be another critera to consider, now that the vegetation's back in the Via Gellia!
My suggestions would be:
1) Devonshire Cavern, Matlock Bath. Entrance may require an adjustable spanner, easiest bit to do is going updip on the old showcave route, having a copy of a decent survey is advisable if you're going downdip plus there's more crawling down that end
2) Groaning Tor level, near Cromford, short walk from Via Gellia Mills. Obvious adit close to a footpath, water should be under welly depth except at the far end after a collapse.
3) Mouldridge Mine, Pikehall, short walk from A5012. Catflap gate leading to walking size passages, more crawly bits if you want. Somewhat complex but survey available in PDMHS bulletin 9:2 on the society's website
4) Hanging Flat Mine, Stoney Middleton Dale, close to Horseshoe Quarry. Collapsed entrance with hole at the back right by the side of the A623, walking height passages.
Follow the horses, Johnny my laddie, follow the horses canny lad-oh!