The Spireites were hit by a three-goal first half salvo and failed to sufficiently recover after the break, despite Hird’s headed goal.
Although the centre-half admitted individual errors played a part in all three of the Lions’ goals, he wasn’t pointing fingers in a public setting.
“We’re really disappointed, frustrated,” he said.
“Everybody is pretty angry to be honest.
“Up to the first goal we’ve played pretty well, we’re in control and then it’s gone downhill from there.
“I could run through the goals individually but I’m not going to do that, that’s a job for us to do as a team behind the scenes.
“We’ve made individual errors today, it’s cost us. It’s our job as a team to make sure everyone is switched on.”
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Millwall’s clinical finishing was in stark contrast to the Spireites’ wastefulness in front of goal, having created enough chances to at least gain a point.
Hird said: “They’ve not had many chances but they’ve punished us for the mistakes we’ve made.
“We lacked that cutting edge again in the opponents’ box.”
Hird insists it’s vital now for Chesterfield to bounce back quickly.
“Everyone is angry because we felt confident coming into this game, we’ve played some good football in every game this season.
“It’s just not good enough.
“We had a good start in the league, we’re on seven points which is nothing to be disgraced about but we need to move forward to Wimbledon and make sure we get something out of the game.”