As well as changing who can book tests, examiners from the Ministry of Defence (MoD) will be brought into the Driver and Vehicle Standards Agency (DVSA) to try to reduce the backlog.
The Department for Transport said 36 MoD driving examiners will be brought in to conduct tests one day a week for a year, at an estimated cost of £100,000.
These examiners are civil servants, not military personnel.
The DVSA has recruited 316 new examiners, but Alexander said that has resulted in a net gain of only 40 as others have left.
Driving examiners will be offered a “retention payment” of £5,000 from next year to try and keep them in the role.
She also said a limit would be placed on the number of times a driver can move or swap a test, and the area they can move a test to once they’ve booked it.
In April, BBC Verify found that the average waiting time across the 319 driving test centres in Great Britain was 22 weeks, with three-quarters of centres hitting the maximum average wait time of 24 weeks.
Every week, the Driver and Vehicle Standards Agency (DVSA) makes more tests available but many are rapidly booked up by bots – automated software that can act much faster than humans.
Companies then resell the test slots to desperate candidates at inflated prices.
The driving test costs £62 for a weekday slot or £75 for evenings and weekends.
But some third-party sites are charging up to £200.