My WiFi signal is really bad in some areas of my house.
I have a spare router and was wondering if it's possible to use it connected with Ethernet, and not directly connected to the modem?
Yes it is, just be sure not to configure it as a DHCP server...
Option 1. Use the router as accesspoint - this way is the best, if your router had the option. this minimizes the processing load on the router.
Option 2. Use the router as "router" - just set it into a different subnet and use the upstream router just as you use the modem. The WAN set as DHCP and remember to set the DHCP server on the router onto a different subnet.
No matter what you do:
First optimize the wireless network - check which channel is least used, and then use the channel. Remember 802.11 channels do overlap. and set different routers into different channel unless you are doing wireless bridging.
And then consider using gain antennae instead of getting another router into the network.
I think this article over at smallnetbuilder.com goes over the steps to do what I think you said you want to do. Hope it helps.
This works, but ideally you would set the device as an accesspoint rather than a gateway. This avoids an extra hop, simplifies subnetting, and can avoid a double NAT depending on configuration. This setting is device dependent, but most products have this option.