There are two other ways, and you can farm mission 7 for Tetradic Tiaras and Tetradic Crowns.
Dismanteling a Tiara gives you Sprint Shoes. Dismantling the Tetradic Crown gives you Hermes Sandals, which can upgrade to Sprint Shoes.
Compared to upgrading weapons, this whole process is very cheap too. While you're at it you may as well upgrade your ATB scarves to Aurora Scarf status, granding a full ATB gauge at battle start plus. Equip both the Sprint Shoes and the Aurora Scarf for starting the battle with haste, a full ATB bar at battle start, and a 10% increase in ATB speed! You can even buy Speed Sashes which you can turn into Aurora Scarves from Moogleworks once you have access to that shop!