This type of hook is not supported by Git. It falls outside the valid reasons for a Git hookvalid reasons for a Git hook as given by Git's maintainer.
The introductory remark in the above linked message speaks almost directly to your case:
I do not particularly like hooks that act after an operation is initiated locally and act solely on local data. This is maybe because I still consider git tools building blocks suitable for higher level scripting more than other people do.
P.S. A “Single Push” Hint
- There are too many caveats for a full explanation, but if you can figure it all out you should be able to deal with the details.
An extra pushurl to a local repo with an “alternates” objects store could give you a low overhead way to locally execute a push hook. But really, the effort is much more than git push upstream && update-web-server (maybe in a shell alias, git alias, or script).