J.W.
Relax. I and my kids have been bitten numerous times by spiders. A teacher of one of the kids did get bitten by a brown recluse but most spider bites are relatively benign. They itch a lot and yes, they can become infected, but even that is fairly rare. Sometimes a person has a bad allergic reaction, and that needs attention.
One of my children always swells up a lot from spider bites (and mosquito bites too), but still, only needed medical attention once after the spider bite swelled up over a quarter of her lower leg (she was small) - and she needed antibiotics. But that was once out of dozens of bites.
If it looks no worse than two days ago - it is doing well. If it gets worse, then go.