You could create a network-of-networks, using the WP Multi Network Plugin. The primary network would be a subdomain network, and the subsidiary networks would be subdirectory networks.
Alternately, you could simply set up a subdomain network, and then use a static front page in each of the network sites, with the blog posts index set to be displayed on a static page named “Blog”. Then, you could modify your Theme, using the home.php template file, to customize the appearance/display of the blog posts index (or, use page.php to customize the appearance/display of static Pages, which might be easier).
But, this somewhat confuses me:
I have a WordPress Network install with the individual sites installed on subdomains. …I would now like to launch a blog on each of the sites. It’d like the blog to be situated in a /blog sub-directory of each of the subdomains…
So, you’re already running WordPress on your subdomain sites, but want to run WordPress on a separate subdirectory beneath each of those subdomains?
On your current WordPress subdomain sites, do you actually have content? If so, are you using static pages to generate that content? If so, then I would definitely go with the second option.