I'll add the Easycome night – every Wednesday at the Old Nun's Head in (unsurprisingly) Nunhead. Not a sessions place, though it does feature a lot of traditional folk music. Tends to lean more towards american folk/bluegrass/Americana. Always busy and well worth a visit. You can just turn up and play if you're prepared to wait.

I'll also recommend the Green Note cafe in Camden. Obviously it's nowhere near Elephant but it's but a short trip up the Northern line. They have a great programme of upcoming gigs and an open mic session on Sunday afternoons.

Given your location, you should maybe also check out The Gladstone pub on Lant St in Borough. That's SE1. You could probably walk it. Again, hardly traditional music as such but in the same neck of the woods - rather more of a psych-folk Green Man festival vibe – but live music most nights of the week.