Caches are instrumental in reducing network and server loads that are typically deployed at the network edge to shorten the response time. Cache optimization for modern delay-sensitive applications is not only concerned with maximizing the cache hit ratio but is also related to minimizing the cache response time. In this paper, we revisit collapsed-forwarding caches and propose COLFT, which is a cache-friendly transport equipped with a cache admission filter that takes into account the download time between the cache and content server. To analyze the performance of COLFT, we developed an in-depth analysis of the cache system that characterizes its behavior, either with or without collapsed forwarding-enhanced cache admission, capturing the effect of the bandwidth share in the transport layer. A numerical evaluation agrees well with the analytical results and shows a significant improvement of COLFT in both the hit rate and response time in realistic scenarios when the transport-dependent download time is considered.