I am going to bet that your Linksys router is an old old WRT54g (blue with black trim) that has 2 antennas on the back. These where the golden years of Linksys and they have taken a step drop off the quality/performance map since then.
Technology changes quickly, so choosing a brand based off of what was good 10 years ago is not the best strategy.
Getting reliable wifi signal in a multi-story house can be difficult to do, even more so if the walls are not just un-insulated 1/2" drywall.
Not sure if by phone line you mean an Ethernet cable (8 pin cable whose plug is wider then a phone) or you mean you get your internet via DSL over phone line and was talking about relocating the router/modem to the 1st story.
You have 2 options at this point:
1) accept that you need 2 medium-end routers like the TP-Link WDR3600 and connect the two of them together with what is called a powerline adapter (this sends data over your AC power wires). The router one not hooked up to the modem will have to be configured as an "Access Point". While this is not that complicated, if you don't understand anything about networking you will need someone to help you set this up.
2) Buy a $150 router like the Asus AC68u and hope it can penetrate your walls/ceilings.