We analyze the performance of enhanced spread spectrum Aloha (E-SSA) in the framework of unsourced multiple access (UMAC). The asynchronous, unframed transmission of E-SSA is modified to enable a direct comparison with framed UMAC schemes, as well as with the Polyanskiy's achievability bound. The design of E-SSA is tailored to the peculiarities of the UMAC setting, resorting to short polar codes and the use of a timing channel to improve the energy efficiency of the protocol. We assess the impact of the preamble length and of the spreading factor on the system efficiency. The resulting scheme exhibits simplicity at the transmitter and linear complexity with respect to the number of active users at the receiver, approaching the UMAC achievability bound in close competition with the best known UMAC schemes.