You don't say what redirection method you're using at the moment, so it's hard to suggest alternatives. I'm guessing, since your old site URL was a Demon sub-domain, that it's a service provided by Demon itself.

If that's the case, the re-direct will be a script on the index page, and will only work from there; the other pages have presumably been removed from Demon's server. That oughtn't to matter much, as the error page that comes up if you try another page specifically advises people to try the homepage and provides a link to it; that is picked up by the re-direct script and you wind up at the new site.

Of course, if people don't bother to try the link, there isn't much you can do about it. The best plan is always to register your own domain name independently (avoid registration/hosting packages; the Cyberhymnal found that that is frequently a false economy), so you can point it anywhere you like no matter how often you change host. You know that now, of course. Still, as I say, not much to worry about really. You may lose a bit of traffic from people who can't read, I suppose.