There are two speeds involved in getting data from a space probe to earth.
Transmission travel time is fixed by the speed of light and being in a vacuum is already at its limit. New Horizons is far enough away that even at the speed of light it takes hours for any signal to reach earth. And any relay stations will just increase that time because of longer paths and processing.
The second speed is the one most impacting New Horizons data return and refers to the number of bits that can be moved per second. This is driven by the ability of the receiving station to detect the signal against background noise over the distance.
So the date rate is determined by the transmitter power and antenna size, the receiver antenna and sensitivity, the distance and the background noise, none of these offer options to boost by 100 or 1000 times on current technology.
Relay stations would help, but they would need to be at least 100 meters across, both to have a sufficiently large antenna and to gather enough solar power to drive a strong signal across the next leg. This would make them larger by a fair margin than the ISS, and costing similar amounts of money to build and send into suitable orbits.
There would also need to be at least three of them, since physics means that a relay between Earth and New Horizons would be moving in an orbit slower than Earth but much faster than New horizons(at least radially) so a relay of relay stations would be needed to get any benefit.
This makes the cost more than is currently justifiable, since the current method works for the amount of data current probes generate.
One potential method to increase data rate is LASER light, since this allows much higher sending and receiving gain, and lower background noise in sensible size/weight. This does require confidence that a sufficiently powerful LASER can operate for the decades so is still in development.