Miller’s injury is not serious but he will not be risked at Gresty Road as the Millers try to progress to the quarter-finals of the lower-league cup competition.
Warne instead plans to use him in the FA Cup game at QPR on Saturday.
"He's just tweaked his hammy a little bit," Warne said. "It had been raining solidly for four days or so and we trained on heavy pitches.
"We had a few aches and pains. Wes (Harding) had a little bit of a groin issue. It wasn't enough to not allow him to play but he felt it in training on Thursday.
"Chieo (Ogbene) was a bit hammy sore, so he had to drop out of training. We were fortunate to get to the Bolton game with only Mickel unavailable.
"It's a thin line. You need the players to train to be ready for matches but if you train in soft conditions you cause problems.
“Mickel won't be involved in the Crewe game. He may be available to play at QPR and if he is he'll play.”
The Millers turn their attention away from League One action, where they returned to the top of the table against Bolton.
Warne said of his side’s performance: “I thought we were good but not at our scintillating best. I take responsibility a little bit.
“I think I messed up their training a bit this week. I think I put too much into him. There was no midweek game so on Wednesday we played an hour on our training pitch.
“We could have had dolphins on the training pitch it was that wet and muddy. The New York pitch is struggling a little bit as well, I think. I trained them Wednesday, Thursday, Friday, which I regretted straightaway.
“However, they still managed to put in a performance, I thought we were good. We put some good crosses in but we weren't as good on our set-pieces.
“I don't think anyone could complain that we won.”