There are 589 potential votes. Eastwatch and Shadow Tower votes are held by their commanders. Castle Black's garrison is under 400 men, so a little over 200 votes are held by Pyke and Mallister. They got 372 votes between them, so they got around 150 votes from Castle Black.
Slynt had 137, with a slow increase over time due to Thorne and Marsh dropping out and backing him. Yarwyck had 72 votes on his own, and dropped out without backing Slynt. In all likelyhood, most of his votes would have gone to Pyke and Mallister.
I think the best option would have been Ser Glendon Hewett. Even with how Martin set up the scene for Jon to show up and pick up the gig, if the characters were allowed to make rational decisions for two seconds, I can see them coming to some sort of arrangement before the night is out.
Pyke and Mallister don't like each other, but don't want Slynt. Ser Glendon in one of the handfull of knights in the Night's Watch, is Pyke's second in command already back in Eastwatch. Jon dislikes the man for waking him up with a kick to the ribs and would have opted to place one of his own cronies there, but still considers him a seasoned brother of the NW. This was when Glendon was escorting Slynt to Castle Black, and he took part in the defense of the Wall alongside Slynt in the time when Jon was in the hole. He is the closest thing to a friend that Thorne can claim to have on the Wall, and comes from a Reach house. He ticks all the boxes to fit the position on account of merit, and doesn't step on any landmines politically. Both in the NW and in relations to the kings in King's Landing and outside the guarded doors.
With Pyke's and Masllister's tokens, he'll already start with more votes than Slynt has, and he is everything that Jon could have ever hoped to be and more.