The action by the Aslef union means members will refuse to do any overtime outside of contracted hours from Monday to Saturday.
Services popular with commuters will be affected, with Thameslink saying it will be forced to run fewer services from May 15 to May 21. The ban will also be in place on Thursday, June 1.
Train operating companies affected include those running commuter routes such as Southern, Southeastern and South Western Railway.
It comes after Aslef members staged a walkout on Friday, bringing train services to a standstill.
Further strikes across 15 rail operators are planned for Wednesday, May 31 and Saturday, June 3.
London Underground lines will not be directly affected.
The union says it is taking the action over “the failure of management to offer a fair deal on pay”.
Announcing the latest strikes, it said: “Most of the drivers have not had a pay increase at all since 2019 and with inflation still well over 10% and the cost of living spiralling, this is not acceptable.”
In an interview on Monday, rail minister Huw Merriman defended the Government’s stance on strikes.
Mr Merriman, in an interview with The House magazine, said that giving in to unions’ demands would create a poor example for other workers.
He said: “If you settle at a much larger figure, then that creates a precedent across the public sector as a whole.
“That impacts inflation, that damages the economy, so you can end up with, settling – which may look on the face of it, would be cheaper to do – [but] ends up having a much bigger financial impact, not just on the railway, but on the economy at large.”
Separately on Saturday, members of the RMT union staged a walkout across 14 rail companies.
Rail passengers faced travel disruption on the day of the Eurovision Song Contest final but the union denied targeting that date on purpose for the strike.