Vermiglio, 36, took over the reins from Matt Jansen in the summer.
Jansen had guided the Magpies to back-to-back play-off defeats after taking over from his ex-Blackburn team-mate Garry Flitcroft in 2016.
It was Flitcroft who gave then player Vermiglio the chance to move into coaching upon his retirement due to injury in 2014.
Vermiglio says his role has developed over time, with Jansen giving him and Smith opportunities.
And he praised Smith for his role in recruitment over the last couple of seasons, work that has been key to helping them guide the Magpies to the top of the table.
After an unbeaten start they have slipped to back-to-back defeats at Bradford Park Avenue (1-0) and Stockport (3-0) on Tuesday night.
But they are still unbeaten at Victory Park as they prepare to host Spennymoor Town on Saturday.
Vermiglio also hailed the final component of their management trio ex-Southport manager and his former Northwich Victoria boss Andy Preece as the best signing of the summer.
“Matt Jansen gave me a good opportunity at the club,” said Vermiglio, who combines his role as manager with his job as a primary school headteacher in Warrington.
“He let me do a little bit of the player recruitment, he let me do a lot of the training, a lot of the tactics and analysis along with Jonathan Smith.
“The key to it is about having the good people around me. Jonathan Smith does an awful lot. He is a bit of an unsung hero when it comes to how well we perform – not just this season but over the last two or three or so years.
“He goes out on a Tuesday night, travelling the length and breadth of the country watching a lot of teams, watching a lot of players and is responsible for some of the good signings we have had.
“We have also had Andy Preece come in. A few people might have raised their eyebrows at that appointment at the start of the year.
“Without being disrespectful to Andy, he was a manager who has been there, done that, certainly the best manager I have ever played for and he was on the hunt to get back in and wanted a job.
“When I approached him I did not think he would want the position. I’m quite proud of the decision that I made.
“I don’t feel threatened at all by him, he is one of the lads, one of the boys, a massive part of this team and a massive part of the success we are putting together.
“Me and Smudger felt that we needed someone else in someone to give us a bit of a fresh outlook because despite keeping the status quo at the club, it is also nice to have some fresh ideas.”