A differentially private selection algorithm outputs from a finite set the item that approximately maximizes a data-dependent quality function. The most widely adopted mechanisms tackling this task are the pioneering exponential mechanism and permute-and-flip, which can offer utility improvements of up to a factor of two over the exponential mechanism. This work introduces a new differentially private mechanism for private selection and conducts theoretical and empirical comparisons with the above mechanisms. For reasonably common scenarios, our mechanism can provide utility improvements of factors significantly larger than two over the exponential and permute-and-flip mechanisms. Because the utility can deteriorate in niche scenarios, we recommend our mechanism to analysts who can tolerate lower utility for some datasets.