The group is buying OHP for an initial fee of £1.5 million and will take on about £1.7m in debt.
OHP employs a team of 25 and also includes Rail Hire UK, which offers overhead line equipment for hire in the rail sector.
The move comes after Speedy Hire jolted the market in September with its second profit warning in as many months.
Speedy hire said OHP generated sales of £2.6m for the year to the end of November, while earnings came in at £800,000.
It said the acquisition is expected to boost earnings within the next year, adding that the cost of the purchase could rise up to a further £800,000, depending on the future performance of OHP.
Chief executive Russell Down said: “I am delighted to announce this acquisition which will help expand our business in this specialist sector of a growing UK rail market and further enhance our market position.”
Analysts at Investec said: “This looks a sensible deal for the group, expanding its offering into what looks to be a growth sector within UK infrastructure.”