Trace-width calculation of a relay output for a 6A device
Your calculator seems a little conservative.  It's closer to the results I'm getting for an internal trace (9.25-9.5mm) than for external traces (3.5-3.7mm).  Not sure which rules it's using, but I've got 5 sites that agree fairly well.
4pcb
7pcb
Digi-Key
Qorvo
DeSmith