In this paper, we revisit the recent small characteristic discrete logarithm algorithms. We show that a simplified description of the algorithm, together with some additional ideas, permits to obtain an improved complexity for the polynomial time precomputation that arises during the discrete logarithm computation. With our new improvements, this is reduced to $O(q^6)$, where $q$ is the cardinality of the basefield we are considering. This should be compared to the best currently documented complexity for this part, namely $O(q^7)$. With our simplified setting, the complexity of the precomputation in the general case becomes similar to the complexity known for Kummer (or twisted Kummer) extensions.