There are many Spotify apps including Pitchfork and Rolling Stone which show a play button in the lower left corner when hovering over an album cover. Here's an example. Does the Spotify API support this natively? What is the best way to code this? Thanks.
It's the standard Player
view. Create it with a track and album URI and insert it into your DOM as you would any other UI element in JavaScript.
Documentation here.
There's an example of how to use it in the Spotify tutorial app, available here on GitHub, in the "Play a single track" example. If you modify that example to use an album instead of a single track playlist, you'll have what you need.