As a SaaS provider, one of the key indicators of a customer at risk of churn is the presence of another competitor in their account. A service which notifies you once an account signs up for a competing service would be immensely valuable in helping to target retention activities.
1) how about you find ways to make retention a continuous objective
2) get used to the fact that not all customers will be retained. I try stuff all the time that I have no idea if it will solve my specific problem. If you do #1 right, I would know that you want to hear about my specific problems and offer ideas on how your saas can solve them. That's the most you can hope for if your service doesn't immediately do what I initially hoped it would (or is more burdensome to implement that I thought, etc)
I can think of a solution that's VERY VERY anti-consumerish.
Being a SaaS provider, create an extensions that you have to install (something consumers want...). This extension should have the permissions to read history / or urls visiting.
Have a blacklist transmitted of urls (so you don't need to transfer the user's data back to the server) and match with it. If it matches...you have to work on that consumer a lot more.
You could employ some sneaky tactics to ferret out if a customer's browser visits a competitor's website... without letting the competitors or the custom know.
This sounds like a massive invasion of privacy, to me. It also sounds like a great opportunity to drive customers further away when you contact them based on that information.