Depends if its a work for hire arrangement.
The issue at hand is, are they paying for your creative design work or are they “leasing” your designs that you retain ownership of for their use?
If the latter, they technically owe you for each iteration based on the template you designed (as if it was a song you copyrighted). If the former, you get paid once and they own it to use however they like, and they own the copyrights (like an advertising jingle).
I don’t think you have a strong case to hold out for payments every time they use the template without you, you’ll chase away most clients with that, so probably the practical thing to do is to charge a bit more up front for the one-time, use-it-any-way-you-want fee, and let them own it all after they pay you once. Be sure to retain the right to reference the original template work in your portfolio, and on web pages, so you can get more work.
Then, as the guy who knows the template best, offer them a discounted rate to adapt the existing work the same way they were going to do themselves, for a lower fee since the hard part is done and you’re just dropping in and tweaking new text and pictures. Suggest a “retainer fee” to “maintain” the project for them for some flat price for three alternate versions, for one year. They might like being able to just forget about it and know that the guy that created the thing will take good care of the changes for them. In any case they are not going to go for paying full-price for each iteration, but may go for a lower rate to, as I said, “service” the template for multiple uses.
Reimagining your situation as a video project, say, a 30-second spot, my clients normally own the whole thing once done and paid for. If they want to do alternate versions, they can take it anywhere or do it themselves, but hopefully they’ll bring it back to me because I already am deeply intimate with the material and their needs. And I’ll bill them each time they bring it back, based on what the project needs. I think this actually brings in more business than holding onto the rights ourselves, which we can’t really do in our situation anyhow. Pretty much everything we do is on a work for hire basis.
It is common, I know, to feel very proprietary and nostalgic about your creative work; your art or your script or jingle or whatever is “your baby”. Let it go. Let me suggest that your creativity is not a finite resource, that you will come up with more good ideas as long as you live, and you should get people to hire you for that, as someone who generates new things of value all the time, rather than trying to squeeze every drop out of your individual old projects done for one-shot products, living off past glories.
This is the difference between a commercial art mentality and a fine arts mentality. Commercial artists give up the stuff for a onetime fee and share it with the world, albeit usually in anonymity or at least, obscurity (except within the trade)… fine artists keep it all close to the vest and use galleries and such to sell it, and may not get paid or recognized until after death. Both can become immortalized, just by different routes.
Not to say you should not get paid fairly and well in any case. But not every job is the Sistine Chapel Ceiling, you know? And if you made something really that good, don’t you think people are going to come looking for you to make more good stuff like what they saw anyway? Are you giving the art away or is it advertising that brings you yet more work? probably both.