Is there an automated way to create another multisite inside of an existing multisite Drupal install?
If you configure multisite instances to use https://www.drupal.org/project/dslm, you can use a command like 'drush dslm-new mynewsite drupal-7.34' to provision a new site.  @btopro has taken DSLM a step further calling DSLM and Drush commands from the CIS install profile to completely automate the process from within Drupal for Penn State and the growing number of schools using ELMS.  This configuration allows schools to generate several new Drupal sites from different ELMS install profiles from a centralized Drupal site.  [This screencast][1] explains how the ELMS CIS implementation works.  There is also a vagrant version of this configuration you can spin up from https://github.com/btopro/elmsln-vagrant

CU Boulder also uses a Drupal front end to DSLM to spin up new sites, but we don't use a true multisite configuration and our backend for this service is more complicated
and CU specific leveraging Eve (http://python-eve.org) and F5 server
(https://f5.com/products/modules/global-traffic-manager).  

I doubt either of these solutions is a good fit for you out of the box, but if the "automated way" you are only looking for only needs to automated this process at a commandline level, DSLM will get you most of what you need. You can also use code from the ELMS CIS project to create your own front end.

If you looking for a graphical front end that works "out of the box", you might want to consider using the Aegir Project (http://www.aegirproject.org/) with an install profiles like Hostmaster (https://www.drupal.org/project/hostmaster).  Installing Aegir on your own server can be complicated (http://community.aegirproject.org/installing/manual), but there are several hosted Aegir options.  Converting an existing site to Aegir is also more complicated than DSLM.  Aegir is really an 
"all in decision" while DSLM can be used along side existing, non-DSLM sites and includes a command to convert an existing site to use the DSLM symlink structure.