You don't need to run ahead. When you enter an area of a park later in the day (e.g., Adventureland in the Magic Kingdom), pick up a fastpass for the closest headliner (e.g., Jungle Cruise), and then "kill time" until it's time to use it, by visiting the nearby attractions with less of a line (Aladdin's Magic Carpets, Tiki Room, Pirates, Swiss Family Treehouse) until it's time to use the Fastpass. Almost all of the major attractions have a few nearby that are far less traveled. For Splash and Big Thunder Mtn., there's the Country Bears, Tom Sawyer Island, the Hall of Presidents and Haunted Mansion... etc.
The only exceptions would be attractions that, by fairly early in the day, would already be giving you Fastpasses with return times several hours away (e.g., Toy Story Midway Mania, Soarin', Peter Pan), meaning you would have to "run ahead" to get a Fastpass if you wanted to use it within a reasonable time. Those attractions are the ones you'll want to plan to visit first thing after park opening, or last thing before closing.