2 ROUTERS hooked up to 1 MODEM? possible?

hi, i was wondering if i could hook up my new router to the modem with the old router already connected to it. the reason im trying to do this is my second xbox wont work with the old router, but there is like 8 devices that rely on the old router. so i don't wana take out the old router. i just wanted to see if u can connect 2 Ethernet cords to the the back of the MODEM? it only has one spot to put a ethernet cord in so would it work with a ethernet cord splitter like this one?>

https://ecatalog.siemon.com/#/en/Work-Area

^^(website)-Y-adapters are available as "splitters" which convert one 4-pair jack into two jacks. The Y-adapters utilize Siemon’s patented UP-2468 plug which allows adapters to be used in 6- or 8-position jacks. The adapter body can be rotated 180° to view either the colored icons or the Y-adapter pinouts, which are printed on the opposite side.

would that be my solution to hook up my xbox to the new router connected to the splitter?

if you could help that would be great. im just trying to get XboX Live work in my room.