We propose a model for optimizing the last-mile delivery of n packages, from a distribution center to their final recipients, using a strategy that combines the use of ride-sharing platforms (e.g., Uber or Lyft) with traditional in-house van delivery systems. The main objective is to compute the optimal reward offered to private drivers for each of the n packages, such that the total expected cost of delivering all packages is minimized. Our technical approach is based on the formulation of a discrete sequential packing problem, where bundles of packages are picked up from the warehouse at random times during the interval [0, T]. Our theoretical results include both exact and asymptotic (as $n \to \infty$) expressions for the expected number of packages that will be picked up by time T, and are closely related to the classical Renyi's parking/packing problem.