Hybrid heterogeneous cellular networks (HCNets) with sub-6G and millimeter-wave (mmWave) base stations (BSs) achieves good performance-and-cost tradeoffs. Via noticing the negligible retrieving latency and the potential joint transmission (JT) gain of sub-6G BSs, we propose a novel cache-enabled user association scheme for hybrid HCNets with limited local storage, in which partial sub-6G local storage caches contents with relatively low popularity for higher overall content diversity. To sufficiently exploit the large bandwidth of mmWave BSs, they are endowed with higher association priority. Then, the success probability and meta distribution of achievable rate are studied for the proposed scheme, respectively. We further provide the optimization of caching parameters and the condition for superior performance over the MPC-based scheme. Analytical and numerical results show that the network performance can be improved for larger sub-6G BS cooperation size and the array size at mmWave BSs, while the mmWave BS density has two adverse effect for low-to-medium and high rate threshold. For the proposed scheme, the content diversity in combination with JT gain benefits the success probability. And its performance advantage increases for larger local storage and/or higher association probability with mmWave BSs.