Inverse Mills Ratio for Logit? For $X\sim N(\mu,\sigma^2)$ ,
$$E[X|X>\alpha] = \mu +\sigma \frac{\phi\left(\frac{\alpha-\mu}{\sigma}\right)}{1-\Phi\left(\frac{\alpha-\mu}{\sigma}\right)} $$
Is there an analogous expression for when $X$ is distributed logistically?

Mills Ratio of a continuous random variable is m(x) = (1-F(x))/f(x). This is the inverse of the hazard, used primarily in survival analysis, where x is survival time.
For x a real number the logistic distribution is
F(x) = 1/(1+e(-(x-a)/s))

where a is the mean and s is the standard deviation.
The logistic density is
f (x) = (-F^2)e(-(x-a)/s)(-1/s) = F(x)(1-F(x))/s
so Mills Ratio is a relatively simple expression
m(x) = (1-F(x))/f(x) = s/F(x).
The hazard is
h(x)= F(x)/s.
The logistic distribution or function is one of a family of distributions where f=c*F(1-F), for a constant c, which gives a relatively simple form to Mills Ratio or the hazard. The normal distribution is not part of that family.
