The giant magnetocaloric effect is widely achieved in hexagonal MnMX-based (M = Co or Ni, X = Si or Ge) ferromagnets at their first-order magnetostructural transition. However, the thermal hysteresis and low sensitivity of the magnetostructural transition to the magnetic field inevitably lead to a sizeable irreversibility of the low-field magnetocaloric effect. Here, we show an alternative way to realize a reversible low-field magnetocaloric effect in MnMX-based alloys by taking advantage of the second-order phase transition. With introducing Cu into Co in stoichiometric MnCoGe alloy, the martensitic transition is stabilized at high temperature, while the Curie temperature of the orthorhombic phase is reduced to room temperature. As a result, a second-order magnetic transition with a negligible thermal hysteresis and a large magnetization change can be observed, enabling a reversible magnetocaloric effect. By both calorimetric and direct measurements, a reversible adiabatic temperature change of about 1 K is obtained under a field change of 0-1 T at 304 K, which is larger than that obtained in a first-order magnetostructural transition. To gain a better insight into the origin of these experimental results, first-principles calculations are carried out to characterize the chemical bonds and the magnetic exchange interaction. Our work provides an understanding of the MnCoGe alloy and indicates a feasible route to improve the reversibility of the low-field magnetocaloric effect in the MnMX system.