Is it alright to use tea tree oil on a cold sore?

I have a cold sore at the corner of my mouth. I have been putting tea tree oil on it because it eases the pain and softens the skin around the sore. I was wondering, is it alright to use tea tree oil this close to my mouth? A little bit of it does get into my mouth at times.
Update: In the past I used Antibiotic ointment (neosporin) on them but I was told that was bad to use near the mouth. The neosporin helped ALOT.
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