Working in a very tick infested woods for a summer, I accumulated many tick-bites (before learning the wonders of vinegar solutions for repelling ticks). Most of the tick removals were done at the end of the day in my cabin, where I did my thorough tick-checks. This was recommended to me and worked very well in my experience of over 1 month with 5+ ticks a day:
Apply a little liquid soap to a paper towel or cotton swab (white
makes it easier to see the tick after it comes off generally). Gently
rub the liquid soap-soaked swab on the tick counter-clockwise and
the tick will come right out as if you unscrewed it. Then save it in a sealed container for ID or testing (Lyme disease tests are controversial and unreliable), or destroy it (by fire, flushing, alcohol...not by crushing with your fingers as you don't want to touch it again and they don't crush easily).
The person who told me about this claimed the tick's mouthpart is a corkscrew and this method literally unscrews them, with the soap lubricating, cleaning, and motivating them to release their hold on you. As I said, this worked for me. This person also recommended using a home-made vinegar-based repellent which worked wonders for me, far better than DEET or Permithrin products.
In any case, you want to get the tick off ASAP - no need to panic and rush, but do prioritize it so to avoid extra unnecessary hours of risking disease transfer. Doing at least daily checks on yourself when in areas where you may get tick-bites, and removing them promptly, is important to avoid tick-borne illnesses. If out in the field with no swab or liquid soap, you could try similar alternatives (I had mixed success using hand sanitizer and my finger in the field) or go with the more conventional tweezers or tick-key removal methods, taking care to pinch the tick as close to the mouth as possible and remove the entire tick. If bitten, make note of it and keep a careful watch for symptoms - if infected, the sooner you begin treatment the better.