Generally people use bouncers (BNCs) for this purpose. You can run your own or pay for one on a stable (dedicated) server. PsyBNC is rather popular. If you use one of them, they act as a kind of IRC server themselves which you connect to using your mIRC. The bouncer will always stay connected to the IRC network you tell it to connect to (unless you tell it to disconnect of course), so when you connect to your BNC you'll instantly reconnect to that network. Kind of like turning on your computer screen when the computer is already running. :P