Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum optimization heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA solves optimization problems using a parameterized circuit with (p) layers, with higher (p) leading to better solutions. Existing methods require optimizing (2p) independent parameters which is challenging for large (p). In this work, we present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and the number of coefficients until convergence, our approach enables construction of high-quality schedules for large (p). We demonstrate our method achieves better performance with fewer optimization steps than current approaches on three problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. As an application of our technique, we observe a mild growth of QAOA depth sufficient to solve SK model exactly, a result of independent interest.