Thanks for reaching out with your question. I would love to help you out with that.
So you can definitely use a Nest WiFi setup with your existing router, but you will need to be mindful of possible double NAT situations. Using more than one router on a network will take some special setup. The double NAT explanation is found here: https://bit.ly/2Xvypy7
If you keep in mind the need to avoid having a double NAT situation, the Nest WiFi should do well to extend the reach of your wireless network. If you have any more questions, please let me know.