The Conway-Maxwell-Poisson distribution (https://en.wikipedia.org/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution) has unbounded support (on the right) and can model both under- and over-dispersion (relative to the Poisson) seamlessly through the use of a single parameter. The Poisson is a special case. It can't handle any amount of underdispersion, though, and it is relatively computationally intensive. It is a member of the exponential family of distributions.
R packages exist for both estimation and regression:
https://cran.r-project.org/web/packages/CompGLM/CompGLM.pdf
https://cran.r-project.org/web/packages/COMPoissonReg/COMPoissonReg.pdf
https://cran.r-project.org/web/packages/compoisson/compoisson.pdf
although I haven't used them so cannot make any helpful comments about their relative quality / usefulness!