What I usually do is that instead of putting the string downward from the hole right away, I do one turn on top and then the rest under the string hole. That way the string is tightly locked between the two winding up and down and that help a lot to avoid the string to escape while tuning it
Guitars:
1978 Fender Telecaster Thinline Custom USA, New Nash TL-72 Thinline Telecaster, 1965 Harmony Meteor, H71, 1986 Fender Telecaster Esquire MIJ, New Martin J-41 Special, 1933 National Duolian, 1941, New Eastwood Mandocaster 12 strings
Amps:
Tweed Vibrolux Custom Denis Manlay, 1976 Fender Deluxe Reverb Silverface