Name for a distribution between exponential and gamma?

The density $$f(s)\propto \frac{s}{s+\alpha}e^{-s},\quad s > 0$$ where $$\alpha \ge 0$$ is a parameter, lives between the exponential ($$\alpha=0$$) and $$\Gamma(2,1)$$ ($$\alpha \to \infty$$) distributions. Just curious if this happens to be an example of a more general family of distributions? I do not recognize it as such.

The density function becomes $$f(s) = {\frac {\alpha}{1-2\,{{\rm e}^{\alpha}}{\it Ei} \left( 3,\alpha \right) }}\cdot \frac{s}{s+\alpha} e^{-s}, \quad s>0$$ where $${\it Ei}$$ is the exponential integral.
• @Cristoph Hanck: 1) seems like it requires registration now? 2) the syntax seems to be very strict. Following their first example $E=mc^2$ gives no results, but ${E=mc^2}$ works. Nov 24, 2020 at 11:33