sorry if you've already played any of these
definitely action platformersThe momodora series and guacamelee, which are excellent, guacamelee has a very developed story, maybe too much for me, though i do love it, i like the more minimalistic sorta stuff, it might appeal to you more
and momodora 3 and guacamelee both have controller support, though momodora 1 and 2 don't, i think, but they are free, while the third one is 4 dollars, i think.
insanely twisted shadow planet's story is sorta vague, but it is an excellent metroidvania-ish game, and probably has controller support
http://store.steampowered.com/app/205730/snailiad is one of the most perfect games for me, though it has no controller support, being a webgame
http://www.kongregate.com/games/auriplane/snailiadalso, phoenotopia, also a free webgame, but larger and more polished than many commercial games, has a very good story, if you are like me, you will be hooked from the loading screen onward
http://www.newgrounds.com/portal/view/644696also 2 by hubol which has no story but is one of the best freeware games there is, sort of a tribute to An Untitled Story, but with many different things that easily distinguish it from that
http://www.hubolhubolhubol.com/index.php?id=2also probably no joystick support
not exactly action platformersgyossait which is an surreal shooting thing and mibibli's adventure which is also surreal, but more an epic megaman tribute, both don't really have stories per se, but have awesome atmosphere, they're very different games besides both being weird platformers, but both made with stencyl
http://www.newgrounds.com/portal/view/582867http://ryanmelmoth.itch.io/mibiblisquestthe swapper is a puzzle platformer, you will either like it or love it, it has some super interesting stuff
http://store.steampowered.com/app/231160/?snr=1_7_7_230_150_2knytt stories if you haven't played it, the original levels are perfect
http://nifflas.ni2.se/?page=Knytt+Storiesand within a deep forest(free) and knytt underground(7$, which is a huge bargain)
there are some really good knytt stories custom levels, though you have to do some searching to find ones that really appeal to you because the archive isn't curated, and knytt stories doesn't have any shooting stuff. my favorite custom levels are awful mountain climber restaurant by healy
https://www.glorioustrainwrecks.com/node/3489, rainy day adventure
http://www.glorioustrainwrecks.com/node/3195, which is a glorious trainwrecks collab, all of talps's and farik's levels, and most of the things on this list:
http://auntiepixelante.com/?p=439it doesn't have joystick support, but worth checking out and stuff
also, if you didn't play glum buster, you may enjoy it, it has a vague sorta symbolic story-ish structure and some very interesting little systems and places
http://cosmind.powweb.com/blog/?page_id=35also probably no joystick support
sorry if this is annoying but i love these things