In addition to enlarging the bezel's oval pin holes toward the inner perimeter, gently increasing the curve between the pins on the click ring (adding spring tension) will aid in tightening up the looseness. Short of putting a kink in the spring, you really can't overdo this, as the underside outer wall of the bezel will confine the curves to some maximum extent.
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I tried this last night and it has improved the left/right play massively on my gmt. I was careful not to apply too much pressure on the spring - just squeezing it gently.
Once reassembled it was much tighter, maybe 1mm of play.
Thanks @KJ2020 for the the tip, was almost about to fork out for a new click spring!