Enhancing classical machine learning (ML) algorithms through quantum kernels is a rapidly growing research topic in quantum machine learning (QML). A key challenge in using kernels -both classical and quantum -is that ML workflows involve acquiring new observations, for which new kernel values need to be calculated. Transferring data back-and-forth between where the new observations are generated & a quantum computer incurs a time delay; this delay may exceed the timescales relevant for using the QML algorithm in the first place. In this work, we show quantum kernel matrices can be extended to incorporate new data using a classical (chordal-graph-based) matrix completion algorithm. The minimal sample complexity needed for perfect completion is dependent on matrix rank. We empirically show that (a) quantum kernel matrices can be completed using this algorithm when the minimal sample complexity is met, (b) the error of the completion degrades gracefully in the presence of finite-sampling noise, and (c) the rank of quantum kernel matrices depends weakly on the expressibility of the quantum feature map generating the kernel. Further, on a real-world, industriallyrelevant data set, the completion error behaves gracefully even when the minimal sample complexity is not reached.