The fault occurred on Saturday
Extra London Underground staff were drafted in after a breakdown in the Oyster card system left thousands of people needing replacement cards.
About 65,000 cards were corrupted and stopped working when they were swiped during technical problems on Saturday.
Tube workers handed out replacement cards and to prevent long queues some passengers travelled free of charge.
Transport for London (TfL) said by lunchtime on Monday more than 30,000 affected cards had been replaced.
A TfL spokesman said: "The vast majority of passengers have travelled without any disruption this morning and London Underground staff have minimised the delay to passengers with cards that are not working.
"We are replacing affected cards and there are now less than 35,000 cards that need to be replaced.
"Ticket offices are well stocked and we advise those passengers who have not yet replaced their cards to go to their nearest LU ticket office throughout today."
The Oyster card system was inoperable for at least five hours on Saturday and some 65,000 cards used during that time stopped working or incurred a fine.
All passengers who incurred a maximum fare or fine as a result of the fault will be given an automatic refund from Tuesday.
The cards are used as payment on buses, Tube, tram and the Docklands Light Railway.
Passengers touch the electronic card on the reader on entry or exit to a station, or when entering a bus.
An investigation has been launched into the problem and TfL is expected to release a statement later this week.