Valve have pushed out their recent beta updates to the Steam client for everyone now, this does include the option to force Steam Play on shortcuts you've added for games outside of Steam.
As an example, a game GOG recently gave away free added to Steam to run in Steam Play—it's Linux gaming store inception!
Additionally, this release also includes the addition of gnutls 3 to the Steam Runtime, fixing network connectivity issues in many Steam Play titles
It also fixes a multitude of Linux-specific issues with the Steam client including those pesky zero-byte updates for Steam Play titles, incorrect mouse wheel scrolling offsets, some games weren't properly detected as running, changing settings constantly asking you to restart when not actually needed, there's a fix with host libssl on newer distributions breaking some titles, a crash when purchasing in-game items in Big Picture and so on.
It's really damn good to see much more attention to the little issues being done in the Steam client lately, credit where it's due Valve have been putting in a lot more effort lately and it's showing.
There's plenty more tweaks like the usual updates to Steam Input now supporting the HORI Battle Pad, HORI Wireless Switch Pad, PDP Wired Fight Pad Pro and the PDP Faceoff Wired Pro Controller. Full update notes here.