Important zones to visit after...

[Katana Puffin]

Taked double jump
And before fight Shidra

Sorry for my bad english

[neojb1989]

Only two things I can think of are

Secret area found in Servant's Path near the bottom right of the map. You need Slow Walk and Double Jump

Secret area found in Nerveroot near the bottom right too. You need double jump and pull self here.

[neojb1989]

There is also a Bloodmetal Chunk you can get, if you go to the Desert Watcher boss fight, go right through the Slow walk path, then double jump off and air dash.

There is a bunch of Bloodmetal and consumables in the top part of the WorldPillar that is easier to get with double jump.

Also the jumping/platforming puzzle at the top and left of Worldpillar is easier with double jump (the one that leads to the Shaper weapon and that one armor set)

That's I think all I can think of worth noting.

[Raineh Daze]

There's a secret area in Cenotaph City, up from previous secret area, with a Crimson Ichor, and there's another one over in Weeping Cavity that might not need it but is much easier to get once you do.