We always used to put on our headphones the half-way (only on one ear) so you can hear the real sound better. But tell your engineer to switch of the signal of the open earpiece in order not to hammer the click into the microphone. More of a problem with not so loud sources like voice-recording...
Another trick is (and in these days with a lot of tracks available very easy to achieve) to have a loud click track from a speaker in the same room and later on make a second track with only the click and phase invert the click-only-track against your actual recording. This will eliminate the click! But of course you may not change anything in the room, speaker position or volume of the click for the two recordings... - but sometimes this works well.
Last one - if you don't need the interaction of the feedback of your amp and your instrument, you could split the instrument's signal, go into another room where you get the loud click on a speaker. The original recording in the other room stays save...