We propose a new inner bound on the capacity region of a memoryless multiple-access channel that is governed by a memoryless state that is known strictly causally to the encoders. The new inner bound contains the previous bounds, and we provide an example demonstrating that the inclusion can be strict. A variation on this example is then applied to the case where the channel is governed by two independent state sequences, where each transmitter knows one of the states strictly causally. The example proves that, as conjectured by Li et al., an inner bound that they derived for this scenario can indeed by strictly better than previous bounds.