Originally Posted by
urbanknight
OP's pic seems to show MTB shoes, albeit with minimal tread blocks.
If those aren't enough, maybe go for a fuller sole, which tends to be common on touring or commuter shoes, and sometimes lower end MTB shoes. They will be more flexible and heavier though, so if you don't want that go with Koyote's suggestion and remove your shoes for the stair climb.
Oh, gosh, you’re right! When the OP complained of slippery-soled cycling shoes, I just assumed that they were road shoes. Did not even look at the photo, I guess.
I agree with your suggestions… there are plenty of great SPD-style shoes with good tread that makes walking pretty easy.