Since the G string is so narrow, it can "grab"less than the other strings on the same sized tuner post.
Couple of options, getting to what jack is hinting at:
1)more wraps around the tuning post, or
2) looping the end of te string trough the slot in the tuning post before the wraps start, to get a stronger anchor point.
But if you've already trimmed the excess string, neither of these options might be available to you.
I faced that "But if" situation .... had only about one turn wrapped onto the peg.
I solved it, and it's held up for over a year. I do have the down-the-center-hole
section of leader, plus about one turn, kinda scary by itself, so I drove a rather
small wood screw down the hole after threading up, to further secure the string
onto the peg.
Very important ! !! ! !! You gotta "unsharpen" [desharpen?] the biting edges of
the screw's threading [outer edges] so it doesn't cut thru the string wrap ! ! !
Effectively, you're converting a biting screw into a nonbiting, jam-it-into-place
peg ... but aided by the now "friendlier" threads, you twist it into place instead
of pushing, or hammering, it straight in [as you'd do with a nonthreaded peg].
You don't need to forcibly torque home the screw, since it's only aiding in the
overall friction game, thus not carrying the whole load of retaining the string.
