The Linnets were well organised but were edged out thanks to Liam Mandeville’s second-half strike.
The result leaves the Norfolk club second from bottom of the league and five points from safety.
"We were in a two-horse race and I know we were 9-1 to win it, but I was happy with our application and that's two sides at the top of the table who we have gone toe-to-toe with in the last few weeks," Widdrington told Lynn News.
"I can't take anything other than positive feelings and positive thoughts from how we've performed.
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"We were a little bit patched up and had cobbled a team together and we've lost to a slightly deflected shot, so I was as proud as punch of the players at the end of the game.
"It's no disgrace coming to a huge club like Chesterfield and losing 1-0. Obviously I don't like losing games of football and it's hard to be jubilant, but it's about how you lose and who you're playing that matters to me."
Widdrington took over from Ian Culverhouse last month and is satisfied with what he has seen so far.
"Every rock I'm turning, I haven't seen anything too ugly under it yet," he added.
"We need another couple of new faces, not players who are going to affect the group of 16 but ones who are going to come in and improve the team.”