The Mariners beat Glossop North End 3-0 on Tuesday to ensure they will ply their trade in the top flight of the Northern Premier League next season.
It was a third promotion in as many years for the club under the stewardship of Mr Thompson, and he is targeting three more in the long-term to take the Mariners to the Football League.
The next goal, though, will be to reach the National League North with a promotion from the Northern Premier League as quickly as possible.
Mr Thompson said: “It’s an amazing achievement to have delivered three promotions in a row, and you have to pinch yourself sometimes that it has happened.
“There are so many people to thank, from our players and the management team to our sponsors, staff, volunteers and supporters.
“Now we want to push on again, and promotion will definitely be our aim again next season.
“You can’t set out in anything in life without the objectives of doing well and striving to push on.
“We like to think of ourselves as a professional, honest, humble club, but that doesn’t mean that we lack ambition.
“We want to really have a go at winning promotion again next season.”
Plans are in place for two new stands to be erected behind one of the goals at Mariners Park during the summer.
An average crowd of about 1,500 has been in attendance at home games this season, and Mr Thompson hopes the club will provide its supporters with many more moments to savour.
He added: “I’d like to think that fans of South Shields will look back on this period in 10, 20, 30 years’ time and be talking about what great fun it was and what an exciting experience it was.
“It’s not just about ‘let’s get to the Football League’.
“It’s the whole experience of trying to get there and what it does for our town and the borough.
“It’s about trying to put South Shields and South Tyneside on the map in a little way.
“We want to promote the club, town and borough, and enjoy it as our success continues.
“We have a clear target and it’s difficult, because you don’t want to put undue pressure on the players and management.
“If you don’t have ambition, though, what’s the point?
“We’re going to push on and achieve the very best we can.
“I genuinely believe we can deliver that (Football League), but without the support of our fans, it wouldn’t be possible.”