Jekyll doesn’t serve anything. It simply generates “final” HTML files that you can host as you like.
There’s no reason why you couldn’t build the new pages with Jekyll, deploy to your hosting server, and leave the old pages.
the server that is bundled with jekyll is a development server really - while it may work in a production environment, it is not recommended I don’t think - it is just there to help you while doing development. Apache would be better, though not sure I could point out the reasons why for a single site.