A body, believed to be that of the missing 28-year-old, was discovered by officers near Caverswall on Saturday, Staffordshire Police said.
A 32-year-old man, who had previously been released on bail, was re-arrested on suspicion of murder.
Two other men, aged 28 and 60, have been arrested on suspicion of assisting an offender, the force said.
The trio of men remain in custody.
In a statement released by Staffordshire Police, Eastwood’s family said: “The family of Samantha would like to ask that their privacy is respected at this terrible time.
“We would like to say ‘thank-you’ to all the people who followed Samantha’s story and tried their best to help.”
The midwife was last seen in uniform leaving work at Royal Stoke Hospital at 7.45am eight days earlier on Friday July 27.
The body has now been removed and a post-mortem examination will take place on Sunday.
The scene in Caverswall will remain cordoned off while forensic examinations continue.
Detective Superintendent Simon Duffy said the force’s primary focus was ensuring justice for Eastwood and her family.
He said: “I would like to add my thanks to those of the family, to all those who have helped and supported the investigation over the past week and I would ask that the privacy of the family is now respected at this tragic time.”