"Miniature" holograms are seen a few times in Star Trek, including the hologram of Tasha Yar, which Data had, and the mini-EMH hologram ("Virtuoso").
This technology presumably had some sort of built-in "mobile emitter," which allowed these holograms to operate outside of a holodeck.
I don't understand why the mobile emitter was so amazing to the Voyager crew. The technology existed prior to this discovery, although it was never shown working on a full-size hologram.
Couldn't the technology used for the Tasha Yar hologram simply be adapted for the EMH? Maybe it was impossible for some reason.