Just to be sure that I understand the current connections (line diagram):
ISP --- (coax, DSL, fiber) ----> Modem --- Ethernet cable --->[WAN port] Old Router [LAN port] ---long messy cable -----> [ Wired Network adapter] Desktop PC.
Feel free to edit and correct my line diagram.
Make and model information for Modem and router(s)?
1) Does the old router support Wifi?
2) Changing from wired to wireless is going to reduce network performance and stability (more interference).
3) You do not need another router. All that you would need is a wireless network adapter for your Desktop PC. However, that wireless network adapter will likely need to have antennas to ensure that wireless is at all reliable and stable. What can happen with most wireless network adapters is that their transmission strength may not be enough to maintain a steady wireless connection.
4) You may be able to configure that old router as an Access Point (AP) but that would still require a router ---> AP wired connection.
Depending on the make and model routers, the applicable User Guides/Manuals may show you how the connections and configurations must be set up.
Example link:
https://www.tp-link.com/us/support/faq/1603/
Overall stay with messy cable and do what you can to improve the cosmetics.
And if possible, take a look at the cable. It may be old, substandard, degraded, etc.. Post what is printed along the length of the cable.
Maybe a new round, Cat 5e, all copper, 22-24 AWG could help with appearances and wired performance.