An expanding civilization could rapidly spread through the galaxy, so the absence of extraterrestrial settlement in the solar system implies that such expansionist civilizations do not exist. This argument, often referred to as the Fermi paradox, typically assumes that expansion would proceed uniformly through the galaxy, but not all stellar types may be equally useful for a long-lived civilization. We suggest that low-mass stars, and K-dwarf stars in particular, would be ideal migration locations for civilizations that originate in a G-dwarf system. We use a modified form of the Drake Equation to show that expansion across all low-mass stars could be accomplished in 2 Gyr, which includes waiting time between expansion waves to allow for a close approach of a suitable destination star. This would require interstellar travel capabilities of no more than ~0.3 ly to settle all M-dwarfs and ~2 ly to settle all K-dwarfs. Even more rapid expansion could occur within 2 Myr, with travel requirements of ~10 ly to settle all M-dwarfs and ~50 ly to settle all K-dwarfs. The search for technosignatures in exoplanetary systems can help to place constraints on the presence of such a "low-mass Galactic Club" in the galaxy today.