Thank you to iFixit for the guide and the replacement part! Despite a few little missing details (now specified in the Step comments), the guide was very useful, took me like 2 hours.

To explain the situation a bit, my iPhone 5S got suddenly in Searching mode. I only realized it the morning after when I got to work. I didn't drop my phone overnight or anything. Only thing in mind, I plugged it on my charger... The day after, WiFi and everything else was working fine. When I took the SIM card out... The phone stayed in Searching mode.

Like any other phone, it got drop a few times and had a few bruises. Although, that sudden malfunction was surprising and frustrating.

I looked online and found a few sites mentioning that for an antenna issue, change the lightning port or DFU the phone (which had a risk of bricking the phone if the software wasn't the issue...)

Changed the connector, reinstalled everything and I am happy to say that IT WORKS!!

No more searching, no issue charging or listening to music!!! Thanks