I wanted to install Python 3.10, but that version is not available in the pyenv version list.
It was checked by pyenv install --list.
People suggested to upgrade pyenv, but I do not see help related to updating pyenv.
pyenv isn't really 'installed' in a traditional sense; it's just a Git checkout. All you have to do to update is
cd ~/.pyenv
git pull
That also updates the list of available Python versions.