The migrants on the Overland Trail from Missouri to Oregon and California figured 15 miles a day to be a good day’s travel. Of course, they had the restraints of livestock and draft animals to deal with since the animals had to be watered and fed at regular intervals and given time to graze and sleep in order to stay in working condition. On the push across (I think) the desert west of the Great Salt Lake, 60 miles without stopping for anything was the best way to get across. It took 48 hours of non-stop travel to do it, but that involved resting the draft animals for two or three days afterwards for them to be in shape for the rest of the trip.
The three miles an hour standard is the marching pace for a an infantry unit on roads. Stonewall Jackson, who was famous for march discipline and covering ground in the Shenandoah Valley in 1862, insisted on the military standard of march for 50 minutes, rest for 10 minutes. This is a pretty rigorous standard when you are carrying every thing you need to live, plus a five-foot long musket and ammunition.
The Fifth Army Corps, Union Army of the Potomac, marched 27 miles in about 15 hours in order to get to the Battle of Gettysburg. This was famous marching at the time.
Sherman’s Army covered about 15 miles a day in the March to the Sea, but they were busy ripping the heart out of Georgia as well as covering ground.
Cavalry on roads was expected to cover 30 miles in a day marching from dawn to late afternoon with a break at noon to feed and water the animals, thus the old song:
Oh, the drums they bang
And the cymbals clang,
And this is the way we go,
Its thirty miles a day
On beans and hay
In the Regular Army, Oh.
Three miles an hour is a reasonable pace for walking on roads, but remember the time needed to provide for the creature necessities. Remember that the standard for the Roman Army marching pace assumed paved roads and permanent camps and minimal congestion. That’s why the Romans built the roads in the first place. The more people you try to put on a road at one time the more congestion is going to slow them up. This is why the Fifth Corps march was so remarkable. The Corps consisted of some 10,000 soldiers with ammunition wagons, ambulances, baggage wagons and artillery and made a column some four or five miles long. Breaking 10,000 men out of camp, forming them into marching columns, walking 27 miles and then going back into camp in a long day is a remarkable achievement.