I'm not sure if this would fit within your limitation of "hyperspace technology", but in the Star Trek TNG episode "Where No One Has Gone Before" a being known as The Traveler used his unique ability to alter time and space, combined with the warp drive of the Enterprise, to propel the ship at unheard of speeds. In this sense, it isn't a pure hyperdrive system that was propelling them, but instead a sort of hybrid organic-hyperdrive system where The Traveler himself was a component (The Traveler only performed this feat while the ship's warp drive was engaged, so it doesn't appear he can move the ship at these speeds on his own).
I don't recall how long the high-speed travel sequences took, but it was likely under a minute or so for each. They made a first trip of 2.7 million light years to the M-33 Galaxy and a second trip of 1 billion light years to the edge of the known universe. Assuming 1 minute of travel time, that places their speed for the faster second trip at around 1.5768×1023 m/s, or over 5×1014 times the speed of light.
...although, nothing can beat the "mere nothingth of a second" that it takes to travel with an infinite improbability drive. But that probably doesn't count as an answer because it doesn't involve "mucking about in hyperspace". ;)