The best replacement for logit is probit.

The best alternatives to logit are rated as following:
★★★★★  probit
★★★★  logistic
★★  ols
★★  probit model
★★  probit regression
★★  logistic regression
★  probit stata
★  log
★  xtlogit
★  multinomial logit
★  hazard
  linear probability model
  probit models
  probit difference
  lpm
  tobit
  probit uses
  ordered probit
  multinomial probit

Your alternative for logit is not listed? Suggest!
Speaking about logit, one probably shouldn't forget about fixed effects.

Other search terms related to logit and their ratings are:
★★★★★ fixed effects
You have in mind a companion for logit and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of logit in all years. Click on graph for more.
logit
probit
logistic