If it was a deer tick (very tiny) and this was within the last few days, yes, go and get a preventative dose of doxycycline. If it was a larger tick, nothing to worry about.
I found out firsthand the 24- or 36-hr rule for deer ticks to transmit the bacteria is bogus. I had a tick on for less than 12 hours and came down with symptoms and a rash within a few days. Treated early, it wasn't a big deal, but it was getting nasty very quickly.
This mirrors my experience almost to the T. Tick was on somewhere more than 12 and less than 24 hours. Noticed the rash and a low fever, headache/stiff neck a week later. First 48 hours on the doxy was miserable. But then felt better rapidly.
Since then if I find a tick attached, I call my doctor, who knows my outdoor hobbies well, and he calls in a one time double dose of doxy. If they’re still crawling or just beginning to stick I don’t, but for anything embedded I do, and demand the doxy.
If they don’t give it to you, ask the doctor to be sure to document that you found a deer tick embedded in you, asked for a preventative dose of doxycycline, and they said no, in their file, because that’s what you’re documenting in yours. Hopefully they’ll get the point then.
Even if your insurance won’t cover it, ask your doctor to call the prophylactic doxy dose in anyway, and pay for it out of pocket. Doxy is a very very very old medicine and is not expensive, especially just for one or two doses as is prescribed prophylactically. Well worth a couple bucks to nip a potential case of Lyme, or whatever else those sacks of **** carry, in the bud. Doxy is the usual medicine of choice for most early tick related infections other than Lyme too.