The temporary break follows the discovery of the body of a third man on Wednesday.
Ken Cresswell, 57, and John Shaw, 61, from Rotherham, and Chris Huxtable, 34, from Swansea, have been missing since the collapse in February.
Police said the search would resume when recovery work in the area where the body was found was completed.
The body was taken from the site yesterday but formal identification has not yet taken place.
Emergency services and workers who carried out the search formed a guard of honour as the body was driven from the site in the early hours of the morning.
Steve Hall, Mr Cresswell's son-in-law, said the family had been hoping for a breakthrough for six months.
He added: "There's nothing you can do.
"Obviously we are hoping this is him, and not in a selfish way to the other two families, because I'm sure they're hoping it's them as well."
The body of a fourth man, Michael Collings, 53, from Brotton, Teesside, was previously recovered from the site.
The building was due for demolition when it partially collapsed in February.
The remaining section was brought down using explosives in July.
The search for the two bodies remaining in the ruins of a collapsed building at Didcot Power Station has been halted.