You have a new customer - celebrate!
Now you need to do something. Yes, you! Do something.
Most first-time buyers have tiny windows of responsiveness. I realize this goes against just about everything you've ever been taught, but it is true. Here's one example ... from a successful brand. Look at incremental rebuy rates (i.e. the likelihood of purchasing in month "x" given that the customer has not previously purchased again). Tell me where the two inflection points are.
Go look at your own data - it's gonna look very similar to this.
The new customer is very responsive for a few months - with an inflection point between 2-3 months after a first purchase. Now the customer is less responsive. Notice the inflection point at Month = 12. When a customer approaches his/her "anniversary" of a first purchase, the customer becomes temporarily responsive. That's an inflection point!
I know how you market to customers today. You largely ignore this fact.
Now that you know this fact, how would you change how you market to new customers? Be specific with your recommendations.