A compact platform for cold atoms opens a range of exciting possibilities for portable, robust and accessible quantum sensors. In this work, we report on the development of a cold-atom microwave clock in a small package. Our work utilises the grating magneto-optical trap and high-contrast coherent population trapping in the lin$\perp $lin polarisation scheme. We optically probe the atomic ground-state splitting of cold 87Rb atoms using a Ramsey-like sequence whilst the atoms are in free-fall. We have measured a short-term fractional frequency stability of $5{\times}{10}^{-13}/\sqrt{\tau }$ with a projected quantum projection noise limit at the ${10}^{-13}/\sqrt{\tau }$ level.