Can you interpose an Action Surge Dash between two attacks?

If a fighter has Extra Attack giving them two attacks, and they want to use one attack on one creature, and the next attack on a different creature that is farther than they can move using their normal move speed, can they Action Surge to Dash before using the second part of their Attack action?

For example, suppose the fighter is standing next to one mook. They take the Attack action and with their first attack, they kill the mook. The next mook is 60 feet away, and the fighter has a move speed of 30. The fighter would like to use his Action Surge to Dash up to the second mook, and then finish his Attack action with his second attack on the second mook.

Normally, you can use your movement between attacks. PHB p. 190:

Moving Between Attacks

If you take an action that includes more than one weapon attack, you can break up your movement even further [than just doing some before and some after, as described earlier in the section] by moving between those attacks. For example, a fighter who can make two attacks with the Extra Attack feature and who has a speed of 25 feet could move 10 feet, make an attack, move 15 feet and then attack again.

The fighter’s Action Surge feature says (PHB p. 72)

Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take an additional action on top of your regular action and a possible bonus action.

The Dash Action is described on PHB p. 192:

When you take the Dash action, you gain extra movement for the current turn. This increase equals your speed, after applying any modifiers.

So, on the one hand Dashing adds to your movement for the turn, and you’re allowed to intersperse your movement, which you now have more of, in between your Attack action attacks. However, it’s unclear to me whether “using the Action Surge feature” is a thing which you can do in between parts of your Attack Action.