In the adaptive influence maximization problem, we are given a social network and a budget $k$, and we iteratively select $k$ nodes, called seeds, in order to maximize the expected number of nodes that are reached by an influence cascade that they generate according to a stochastic model for influence diffusion. Differently from the non-adaptive influence maximization problem, where all the seeds must be selected beforehand, here nodes are selected sequentially one by one, and the decision on the $i$th seed is based on the observed cascade produced by the first $i-1$ seeds. We focus on the myopic feedback model, in which we can only observe which neighbors of previously selected seeds have been influenced and on the independent cascade model, where each edge is associated with an independent probability of diffusing influence. Previous works showed that the adaptivity gap is at most $4$, which implies that the non-adaptive greedy algorithm guarantees an approximation factor of $\frac{1}{4}\left(1-\frac{1}{e}\right)$ for the adaptive problem. In this paper, we improve the bounds on both the adaptivity gap and on the approximation factor. We directly analyze the approximation factor of the non-adaptive greedy algorithm, without passing through the adaptivity gap, and show that it is at least $\frac{1}{2}\left(1-\frac{1}{e}\right)$. Therefore, the adaptivity gap is at most $\frac{2e}{e-1}\approx 3.164$. To prove these bounds, we introduce a new approach to relate the greedy non-adaptive algorithm to the adaptive optimum. The new approach does not rely on multi-linear extensions or random walks on optimal decision trees, which are commonly used techniques in the field. We believe that it is of independent interest and may be used to analyze other adaptive optimization problems.