Centre-back Williamson was told he could leave St James' Park in pre-season and looked set to be on his way out of the club.
However, a move never materialised and the 30-year-old has started Newcastle's last nine league games, since appearing in September's 3-2 defeat at Everton as a half-time substitute.
He told The Northern Echo: "A chance came along for me at Everton, and hopefully people feel as though I've taken it the best I could.
"I've had some good times here, but the last few months have probably been the most enjoyable of all.
"I think the manager knows what he's going to get from me - I defend first and try to keep things tight and organised. That's what I'm about.
"Hopefully it's becoming difficult to drop me because we're winning and defending fairly well.
"I've had some good times here, but the last few months have probably been the most enjoyable of all."
