Yes, the general procedure you are describing is a site-to-site transport.
- Site-to-site transports can be initiated by the person needing to be beamed as long as a transporter is assigned to the user in question. The transporter can be set up to automatically initiate the transport upon command.
A site-to-site transport was a special type of transport in which an object or person was transported from one site directly to another, neither site being a transporter platform. A site-to-site transport was accomplished by first transferring the transporter's target from the site of origin to the pattern buffer of the transporter, in the same manner as the usual "beam in" procedure; instead of being routed to a transporter platform, however, the matter stream was diverted to a second site, in a similar procedure to a conventional "beam out". Memory Alpha: Site-to-Site Transport
- Such a transport is time intensive, energy intensive and rarely used except in medical emergencies to beam crew members directly from the site of injury to medical facilities. This process was rarely used for mass transporter use because it effectively halved the speed of transport.
The first known use of this technology: While on Earth in 1986, Montgomery Scott utilized site-to-site transport capabilities on board a Klingon Bird-of-Prey to transport an away team directly from the elevator of a hospital facility to just outside where the ship was parked. Technically, this was the chronological first known utilization of this technology. (Star Trek IV: The Voyage Home)
Away Mission Emergency Transport
The standard emergency transport technology used in the Next Generation was the emergency transporter armband. This device allowed for remote access to a transporter preconfigured to transport without any verbal command.
An emergency transporter armband was a device used by Starfleet personnel during an away team mission to remotely activate the transporters of a starship or runabout for an immediate beam-up in the event of a dangerous situation where verbal communication was not possible, preventing a request for transport in normal circumstances. The armband comprised a type-7 phase discriminator and a subspace emitter. (TNG: "The Best of Both Worlds")
Experimental Transport Technology: The Next Generation
There is also the emergency transport unit technology which was largely experimental during the TNG series of Star Trek.
The emergency transport unit was a portable transporter device developed by Starfleet in 2379. The small device was designed to provide an individual with a single, one-way transport in order to escape a potentially dangerous situation.
The first successful field test of the emergency transport unit occurred during the Battle of the Bassen Rift, when Data used the prototype to beam Jean-Luc Picard to the USS Enterprise-E before destroying the thalaron weapon aboard the Reman warbird Scimitar. (Star Trek Nemesis) REF: Memory Alpha: Emergency transport unit