The news comes a month after 39 ex-post office staff, including a number from the West Midlands, were granted appeals against convictions relating to shortfalls which may have been caused by glitches with the Post Office's computer database.
Rubbina Shaheen, who kept a post office in Shrewsbury, Tracy Felstead, who lives in Telford, Neelam Hussain, a former sub-postmistress in West Bromwich, and Carl Page who kept a branch in Rugeley, are all fighting to clear their names after being jailed for discrepancies they say was caused by a fault with the Horizon computer system.
The offer comes on top of a £57.75 million payout the Post Office made to 555 ex-post office staff – including Miss Felstead – in an out-of-court settlement.
The payout was in settlement of a group litigation in the High Court, which found that previous versions of the Horizon system contained software defects which caused shortfalls in the subpostmasters' branch accounts over a number of years.
It led to hundreds of staff being accused of false accounting and theft, resulting in some being made bankrupt while others were prosecuted and jailed.
Nick Read, group chief executive of the Post Office, said: "We are resolving past events fairly where we got things wrong.
"The launch of this scheme is an important milestone that demonstrates a more open and transparent relationship with postmasters and offers redress for those who may have experienced shortfalls related to previous versions of the computer system Horizon."
The scheme is open to sub-postmasters who were not part of the last year's litigation.
The Post Office said the computer glitches related to previous versions of the Horizon system, and added that the version in use today was reliable.
The scheme is now open to both current and former postmasters and full details, including eligibility criteria and an application form, are available at onepostoffice.co.uk/scheme or by emailing email@example.com Applications should be received by midnight on Friday 14 August 2020 to be eligible.
Mrs Shaheen, 54, was jailed for 12 months in 2010 for false accounting, while Miss Felstead, 37, was jailed for six months in 2001 after being convicted of stealing £11,500.