When it was very windy over the last weekend, my backyard fence was
swaying quite vigorouly. to the extent that if I didn't secure it, it
would've bent or broken. I guess one of the posts wasn't secure enough
inside the ground. so we tied that post to the one opposite to it and
it stood the wind for that day.

I need a long term solution for this. Should I remove that section, dig
the hole deeper for the post, possibly put some concrete in it and
secure it that way? Or is there an alternate solution that works just
as well without removing the section of the fence?

I don't have much help or tools to do this. Could you offer any
suggestions? Thanks.