I have a fair number of links to my homepage but it is taking a while for the SEs to index all my pages. If I get some backlinks to pages that are deep in my site, will the rest of them get indexed faster?
Deep links will help plus ensure that your links aren't hidden in javascripts that the bots aren't able to follow. also add a sitemap to the footer of the page.
In general, the more in-links a page has, the more likely it is to get indexed. So the basic answer to your question is YES. But the links don't have to come from another site. You can also get those pages indexed faster by giving them more links from within your own site, such as linking some of them to each other.