We propose a new coding scheme for the discrete memoryless two-user multi-access channel (MAC) with rate-limited feedback. Our scheme combines ideas from the Venkataramanan-Pradhan scheme for perfect feedback with ideas from the Shaviv-Steinberg scheme for rate-limited feedback. Our achievable region includes the Shaviv-Steinberg achievable region and this inclusion can be strict. For general MACs and for sufficiently large feedback rates, our scheme outperforms the Shaviv-Steinberg scheme as it achieves the same rate region as the Venkataramanan-Pradhan scheme for perfect feedback (which cannot be achieved by the Shaviv-Steinberg scheme). Furthermore, we numerically evaluate our achievable region with a specific (Gaussian) choice of random variables for the memoryless two-user Gaussian MAC. Our simulation results show that for some parameters of the Gaussian MAC and the feedback rate, our scheme achieves a strictly larger sum-rate than the Shaviv-Steinberg scheme.