Injecting gas into the beer out line makes little difference - the size of the bubbles are not small enough to dissolve any significant amount of gas, so they just float to the top and become part of the headspace just as if you'd used the gas in line.
One way to speed up carbonation is to first chill the keg to 41F/5C or cooler, and set the CO2 regulator to 30psi. Connect the gas, and roll the keg back and forth on it's side for a minute. Then disconnect the gas and continue rocking for a further minute or two. The cold beer combined with the rocking motion under fairly high pressure gets the carbon dioxide into solution much quicker. The second rocking without gas helps equalize the pressure. You can then dial the regulator to the target PSI and continue rocking. You stop when you don't hear any more gas entering the keg, usually in just a few minutes.
Leave the keg for a couple of hours and then it's ready for serving.
One downside to this is that you have to either drink the beer quickly (within a week or two max) or take steps to purge the keg of oxygen. The shaking will dissolve any oxygen in the keg into the beer cause it to stale quicker. But it's a winner if you need to carb in a hurry and are sure the beer will be consumed quickly (e.g. a party.)
I've got 4 kegs ready for a party this weekend that I'm carbing up like this.