Doing longer miles per day isn't just about average walking speed; at least as important is hiking "style".
I.e., wake with the sun, and since the OP suggested a hike in the Fall, then wake *before* the sun, eat and pack up in the dark, set foot on the trail as soon as it's light enough to see a couple feet ahead. You particularly want to make the most of all of the daylight hours when daylight is in shorter supply --- not so much an issue in June, more of a factor in early spring or late fall.
Set a truly sustainable pace (really "energy output" as pace varies with trail conditions) and don't take breaks, or at worst take short ones. I think that lunch is a good exception, to rest, eat, perhaps treat water, air out your feet, possibly dry out stuff, plan the second half of the day.
The other piece is just being efficient in camp, but if necessary you can make camp at the end of the day in fading light. In shoulder seasons you've got plenty of dark hours to sleep in.
If you spend most of the daylight hours walking, you don't have to walk particularly fast to do decent mileage, especially in summer.
You do need to be in condition to keep walking all day, however, and of course having a lighter pack helps a lot. Even so, unless you seriously train ahead of time, I suggest that you don't plan on doing 20+ miles the first few days out.
But do the math: if you can average 2 mph and walk for 10 hours in a day (out of 24 total), voila --- 20 miles. That leaves you with 14 hours for breaks, lunch, and in camp in the morning and evening.