Maybe, but you would need a continuous wired ethernet connection from the basement to the third floor. If the house is fully wired for ethernet, then I don't understand why the modem needs to be in the basement. It sounds like the house may not be fully wired. I tried to Google this, but it seems complicated. The internet provider may not allow two routers to send out the same wifi signal.

wi-fi extender
 

cheapest way is with a wi-fi extender. you can select which wi-fi you want to connect to. If it doesn't work always send back to Amazon

Try moving the existing router to somewhere else in the basement.
Then try a different router. Your may be defective. If it's provided by your internet provider, call them and ask for service.
If your router is separate from the internet modem, and the two are connected with an ethernet wire, you can relocate the router to the 3rd floor.

Routers or switches? With the proper router with NAT capabilities, the cable modem Ethernet goes to the router, and from the router to switches/splitters for multiple drops/access points. You can have multiple access points on a NAT router