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.
I just wanted to follow up real fast to see if you were able to get this resolved or if we needed anything more on this. If you are still looking for help, let me know and I'll be happy to keep working with you on this.