Danny Rowe and ex-Town player Jeff King put Chesterfield 2-0 up before substitute Matty Stenson got one back for The Shaymen.
"I think we've had all the ball, dominated the game, we've limited them to half-chances and the two goals we've conceded are disappointing, goals we don't normally concede, which is utterly disappointing," Wild said.
"But for all our endeavour, our qualities, those moments have let us down tonight.
"I thought we were excellent second-half, I thought we moved the ball well, we just didn't create enough chances.
"But we've got a lot of players who've played a lot of games, who've never played that amount of games before, and I think in the last two weeks it's caught up with us."
When asked if he felt Chesterffield had handled the occasion better than Halifax, Wild said: "I don't think it's anything to do with the occasion, I think it actually took a red card incident to liven the game up.
"I thought it was a good game after that little tussle, but we all know the referee has made the wrong decision. There's three of them within five yards and they can't make the right decision, it's just baffling."
Wild added: "I always thought we were going to get something, I never felt like we were never going to get something.
"We were so on top, I just never felt like we were going to lose.
"But I think unfortunately, our quality in those moments that matter has cost us this evening."
Town's cause wasn't helped by an injury to Kian Spence in the warm-up.
"The young lad's played a lot of minutes, but I thought Woodsy came in and did well," Wild said.
"Losing one of our key players, then your bench is different so your plan of when to bring people off the bench has hurt us."
Midfielder Kieran Green lasted less than an hour, and clearly wasn't 100 per cent fit, unable to influence the game in his usual way.
"Greeny did well to get to 55 minutes. Certainly in the midfield areas we're down to bare bones," Wild said.
"I thought Harvey (Gilmour) had another quality game, I thought he was excellent at Stockport last Sunday, and again he's been excellent tonight and he should take great heart from that."
When asked what he had said to his players after the match, Wild said: "I've just said thank you, what a phenomenal season.
"Some of the landmarks we've hit as a football club this year have been unreal.
"You come to work to enjoy it and I've enjoyed ever day of this season, and that's because of the people you work with."
Town attracted their biggest crowd of the season for the game.
"Great atmosphere, I'm just gutted we couldn't do it for them tonight," Wild said.
"But unfortunately I think our race was run. And I think if we'd gone on to the weekend that would have exposed us even more.
"Unfortunately, we are where we are."
Reflecting on the season, Wild felt the players had given everything they possibly could.
"I think we do every week and I think that's probably where we're at," he said.
"We've given every ounce this season. I said to them on Sunday night, and I've reiterated it in there now, that 95 per cent of them have probably hit personal landmarks this season that they've never hit in their careers before.
"So all things considered, I think that's massive."
On next season, Wild said: "It's not for now to talk about next season, I can't my head around that we're out yet, so I need to go home and reflect on that, and then see where that takes us."
And questioned about his own future, the Town boss said: "Again, it's not for now, I need to go home and reflect, sit down with the players and take it from there."