In 2368, the Hornet was part of the task force that blockaded the Klingon-Romulan border during the Klingon Civil War. During the initial deployment of the tachyon detection grid, the Hornet was linked to the USS Ajax, USS Enterprise-D, USS Hood, and USS Tian An Men.
During an attempt to lure the Romulans through the detection grid, Captain Jean-Luc Picard decided that the USS Excalibur should withdraw from the formation with feigned engine troubles, forcing the Hornet and USS Akagi to be repositioned. In doing so, this would open up a small gap in the detection net for the Romulans to attempt to pass through. The ploy, however, failed when the Romulans disrupted the network near the Hornet, forcing it and the rest of the fleet to withdraw and regroup at Gamma Eridon. (TNG: "Redemption II")
In 2375, during the Dominion War, the Hornet put in for repairs at Deep Space 9. However, in order to bring forward repairs of the Romulan warbirds IRW Dividices and IRW Genorex, the Hornet, as well as the IKS Rotarran, had to stand by. (DS9: "Inter Arma Enim Silent Leges")
The ship's registry appeared in the okudagram depicting the tachyon detection network in "Redemption II". According to the Star Trek Encyclopedia, the class of this starship was Renaissance-class. In the first edition of the Encyclopedia, and later corrected, the Hornet was given the registry NCC-10523.
According to the Encyclopedia, the ship was named after the American aircraft carrier USS Hornet (CV-8), which saw action at the Battle of Midway, but was later sunk in October 1942. The named then passed to CV-12, launched in late 1943. Hornet CV-12 also had a link with space exploration, serving as the recovery vessel for the Apollo 11 and Apollo 12 missions.
Ironically in "Redemption II", Starfleet's Hornet served alongside USS Akagi, a ship named after a Japanese carrier that fought against the real life Hornet at Midway - episode Ronald D. Moore thought it fitting that, in a future where Humanity had become united by its own accord, ships whose namesakes had fought one another would now fight side-by-side.
According to the Star Fleet Technical Manual, there was a Constitution-class USS Hornet in service in the 23rd century with the registry NCC-1714 (although the game supplement Federation Ship Recognition Manual gives its as registry NCC-1778). This would seem to be the same Hornet that was mentioned as a combatant in the novel Dreadnought!. According to the DC comic book "Gateway", she was captained by the sister of Edward Hagler.