By far, the best essential oils for bug bites and general owies are lavender and tea tree oil. Always keep these on hand to tackle any mystery bump or bite (diluted in carrier oil, of course).
I became a loyal fan of lavender when I saw a few drops of it heal a red, hot, spreading spider bite on my toddler, a growing skin infection (cellulitis, I believe) from a mosquito bite, and a blood infection that was a black line travelling up someone’s arm from an infected burn. It only took a couple drops, and all of these things went away completely. The blood infection vanished within two hours of applying the oil once and the infected bites within two days after applying it a couple times each day. (This was before I knew to dilute them, though. I strongly believe in diluting them now so that you don’t become sensitized to them.)
And my husband had a bad spider bite on his knee that made his knee hot and swollen and red. We tried various oils on it, but the only one that worked was Tea Tree Oil. It healed it within a few days.
If you only ever buy two oils, make it these two.