Outpatient and midwifery services will move to Alton Community Hospital on March 31, with X-ray services going there at the end of June.
The trust is exploring options to keep paediatric physiotherapy services in Bordon, while the Royal Surrey County Hospital NHS Foundation Trust is now providing community midwifery at the Chase.
Hampshire Hospitals chief executive Alex Whitfield said: "We are sorry for any additional inconvenience these changes may cause for our patients.
"Unfortunately, as we have to hire the space at the hospital and have had declining numbers of patients attending for the small number of clinics we run there, we can no longer afford to provide services from the Chase Community Hospital.
"We have spoken to a wide range of people in the community and have worked very closely with South East Hampshire Clinical Commissioning Group to arrive at this decision.
"We are providing the majority of services at Alton and these services will be run by the same consultant and clinical teams to provide continuity of care."
A potential Chase X-ray patient said her doctor was unaware of the service. Lynda Skinner, posting on the Nextdoor Whitehill and Bordon page, said: "I was somewhat surprised at the reply I received when I asked my doctor in Liphook if I could be sent for an X-ray at the Chase. He said he never thought about using the Chase. Is that because the Chase Hospital does not, or has not, informed local practices of what services it provides?
"We really need the Chase Hospital - but if doctors are not reminded of local amenities, how are they supposed to use them? I ended up driving to Petersfield when I could have walked to the Chase."
Conservative Hampshire County Council member Cllr Adam Carew said: "Having led and co-ordinated our community’s campaign to Save Our Beds, I am utterly appalled that Hampshire Hospitals, which is based around Basingstoke and Winchester hospitals, is withdrawing health services from the Chase when they are quite happy to provide them in Alton.
"These cuts are not welcome or well thought out. They come at a time when our population in Whitehill & Bordon is starting to expand.
"They have cited low attendance and the cost of running the clinics as key reasons for their withdrawal, yet it was made very clear at Hampshire County Council’s health and social care scrutiny committee (HASC) there was no issue with local patients not attending.
"We have also questioned NHS Property Services about their rental charges at the Chase. As the local member, I also made it very clear that although there are buses to Alton, public transport is not great in this area.
"We need to be very clear. These are not national NHS cuts, these are very much a Hampshire Hospitals decision - and as Whitehill & Bordon’s county councillor on HASC, I have spared no quarter in opposing them. HASC, which has only a scrutiny role, has made its views known - but has strongly welcomed the attempts made by our local clinical commissioning group to secure these services elsewhere."
Whitehill & Bordon Community Party leader Cllr Andy Tree said: "I am dubious how the statistics are derived.
"For example, I was asked by a consultant in Guildford to go to Haslemere for blood tests and my wife was told to go to Alton for a scan. We surely then get logged as patients in those towns when in fact we should be logged as patients in Whitehill & Bordon who would use Chase Hospital if offered.
"It is very disturbing that our own NHS trust is moving our midwifery and X-ray services to Alton. How can it be fair that pregnant women who do not drive have to travel to Alton if they want to use their own local NHS trust and have their baby in Basingstoke?
"It is absurd we are having to rely on the neighbouring NHS trust to support us by offering midwifery services in Whitehill & Bordon.
"These changes take away choice from prospective parents."