Is there a shortcut to connect Bluetooth headphones on iOS? Every time I want to use my Bluetooth headphones with my iPhone (iOS 12) I have to open Settings, tap Bluetooth, then select the device. 
Is there a way to connect with less friction?
Confirm that your iPhone Bluetooth is always on

Confirm that you don't have a competing device trying to connect to your headphones

Depending on your headphones, they may have a parameter in which they only auto-connect to the last device.  Do you connect them to anything else?

Experiment a little bit with the connection - What is the exact sequence by which you try to get them to auto-connect?  Maybe you need to unlock your phone before turning on your headphones?  Or try opening whichever app you're using to listen to music (I assume) before turning on your headphones.

If none of that is useful this article on AppleToolBox discusses various Bluetooth troubleshooting specifically written following the iOS 12 update.

Good luck!