Do that..
Years ago, a visiting ill-mannered dog scared my SO's cat under fridge. In trying to retrieve her, P. was bitten deep into hand (she was terrified; otherwise a sweet cat). By that evening there was swelling.. much Much worse: beginning of the tell-tale red striations along arm, of ~'blood poisoning'.
Emergency-room time (this was before they were jammed 24/7.)
Dr. said, "You got here in time; had you waited another 24 hours ... {Baad = listed possibilities.} If it were my family, I'd recommend she stay overnight.."
We settled for this being one of those cases for which antibiotics were meant to be used. Lots of Penicillin-G worked.
YMMV, but a deep puncture wound evades the effective surface containment barriers; toxins circulate, making it impossible for the T-cells to concentrate effectively - on a local area.
Luck