For releases on CPAN, usually the author gives you co-maintainership via PAUSE. That way, you can upload new versions of the module onto PAUSE and CPAN.

The alternative is to bug the original author to make the release, but if they already handed over the module to you, most likely they don't want to deal with the bug reports etc. either.

To further what Corion said, you can see the permissions on the module by clicking on "Show Permissions" link after you've logged into PAUSE, then searching for the original author/module. If you've been granted co-maint, you'll see something like:

module    userid    type    owner
CGI::Application::PageBuilder     CMOORE     first-come     CMOORE
CGI::Application::PageBuilder     STEVEB     co-maint     CMOORE
[download]

...which means you can upload with your account and it'll take precedence over the original authors most recent previous release. "first-come" means the original author, who actually created the namespace. If you haven't been granted co-maintainership and you upload the distribution with your own PAUSE account, the dist will show up as "Unauthorized", which means your version will not be automatically installed via any CPAN client, the previous version by the original author will.

Also, in the dists I've taken over, I always leave the original author's information in the documentation and copyright, and sometimes add my name as a person who has taken over maintenance (if any work I've done is more than a one-or-two line change (I notice that the vast majority of others do the same).

As Corion said, the original author can give you permissions. If he/she is no longer reachable, you'll need to write to the modules mailing list asking for permissions for that module (giving proof that you attempted to contact the original author). See also for example http://neilb.org/2013/07/24/adopt-a-module.html