In the balanced allocations framework, there are \(m\) jobs (balls) to be allocated to \(n\) servers (bins). The goal is to minimize the gap , the difference between the maximum and the average load. In 2015, Peres, Talwar and Wieder used the hyperbolic cosine potential function to analyze the challenging case where \(m\gg n\) , for a large family of processes, including the \((1+\beta)\) -process and graphical balanced allocations. The key ingredient was to prove that the potential drops in every step, i.e., a drift inequality . In this work, we improve the drift inequality so that (i) it is asymptotically tight (leading to tighter gap bounds), (ii) it assumes weaker preconditions (thereby resolving an open problem regarding weighted graphical allocations), (iii) it applies not only to processes allocating to more than one bin in a single step but also (iv) to processes allocating a varying number of balls depending on the sampled bin. Our applications include the aforementioned large family of processes, and also several new processes and settings, including outdated information and memory. We hope that our techniques can be used to analyze further interesting settings and processes.