Gardaí said the man was detained in Co Leitrim on Friday evening.
He was taken to Carrick-On-Shannon Garda Station under Section 30 Offences Against the State Act 1939.
The PSNI also continue to question two men, including prominent republican Colin Duffy, about the killing.
The pair, aged 44 and 31, were detained in the Lurgan area on Friday morning and taken to Antrim Serious Crime Suite for questioning.
Mr Black, a 52-year-old father-of-two from Cookstown, was ambushed by gunmen while driving to work at Maghaberry along the M1 near Lurgan on Thursday morning.
A number of shots were fired at his black Audi A4 (reg HHZ 1560) from a dark blue Toyota Camry with Dublin-registered number plates (reg 94 D 50997), which pulled up alongside.
It is widely believed that dissident republicans are behind the attack as they have been engaged in a long-running protest campaign against conditions in Maghaberry.
Earlier this year, Lurgan man Duffy was acquitted of the murders of two soldiers shot dead by dissident republicans outside Massereene military barracks in Antrim in March 2009.
Meanwhile, a reward of up to £10,000 has been put up for information leading to the arrest and conviction of those responsible for the murder. Independent charity Crimestoppers is behind the reward fund.
"This heinous crime has sent shockwaves through communities across Northern Ireland and left a family without their father," Roger Critchell, director of operations for Crimestoppers, said.
"I am urging anyone with information about this crime, no matter how small that information may be, to contact Crimestoppers anonymously."
We don't want to know who you are - only what you know.
PSNI Assistant Chief Constable for Crime Operations Drew Harris added: "If the reward encourages some individuals to come forward, this will be of benefit to all our efforts as we work to secure convictions against Mr Black's killers."
Police have also launched a fresh, specific appeal for information about the movements of the vehicles belonging to both the victim and the gunmen in the hours before the shooting.
"We need to hear from anyone who saw David Black's Audi A4 at any stage from shortly after 7am yesterday, when it travelled from Cookstown, through Stewartstown and onto the Tamnamore roundabout, before going onto the M1 eastbound," PSNI Superintendent Keith Agnew said.
"We also need to hear from anyone who has seen the car used by the killers in the past few days. This car was found burnt out in the Inglewood area of north Lurgan at 7.50am on Thursday."
Mr Black, a long-standing member of the Orange Order, is the 30th prison officer to be murdered in Northern Ireland since 1974 - though the first for almost 20 years.
The murder of the prison officer has been widely condemned by politicians and leaders, including the First and deputy First Minister, Taoiseach Enda Kenny and Prime Minister David Cameron.
Meanwhile, a book of condolence for Mr Black has been opened at the Burnavon Centre in his hometown of Cookstown by the council's SDLP chairman Tony Quinn.
Mr Quinn said it was the least the community could do to show its respect.
"The book of condolences is there, not only as a tribute to the high esteem David Black was held in the community, but it also gives local people the opportunity to unite and pay their tributes to the senseless killing of one of their own," he added.
"The council is unanimous in its condemnation of David Black's murder."
During times like this, a community has the opportunity to send a stark message out to those who carried out this brutal murder - not in our name, not in the name of the people of Cookstown.
Tony Quinn, SDLP
Ian Simpson from the NI Prison Service Benevolent Fund, who met with Mr Black's family on Thursday night, said it was a "very sad and very moving experience at this tragic, tragic time".
He said that David's widow Yvonne had said she only wanted to know: "Why and why now?"
"The Prison Service family will obviously pull together at this time to be there for the family," he added.
"As Yvonne said, nobody would want anybody to take any retaliation for what happened today and we certainly would not want this to be repeated."
He described the late prison officer as a "very well respected man", who was well liked and dedicated to his job.
"He had served over 30 years in the service and I think everybody in the service who knew him, who worked with him, held him in the highest regard," he said.
"He is somebody who will be sadly missed."
Anyone with information about David Black's murder is asked to contact the special incident room at Lurgan PSNI station on 028 38 315291 or Crimestoppers on 0800 555 111.