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The novel technique proposed in this paper for dynamic cache size tuning is an enhancement of the previous MACSC (Model for Adaptive Cache Size Control) approach. Similar to its MACSC predecessor the Enhanced MACSC (E-MACSC) technique consistently maintains the given cache hit ratio. The focus of the research is presently on supporting small caching systems of limited recyclable memory resources. The MACSC tunes the cache size adaptively with the instantaneous popularity ratio, which is computed statistically by the point-estimate (PE) method on the fly. It is difficult to harness the PE convergence time, because the following are unpredictable: (a) the number of data samples needed by the PE process to achieve convergence and (b) the inter-arrival times among these data samples. In the E-MACSC framework this unpredictability problem is resolved by replacing PE with the M^3RT mechanism, which is a realization of the Convergence Algorithm (CA). Therefore the E-MACSC is also called the MACSC(M^3RT) as compared to the original PE-based MACSC or MACSC(PE). The CA is an IEPM (Internet End-to-End Performance Measurement) technique that measures the mean of a waveform quickly and accurately. The CA prediction accuracy, however, differs from other IEPM techniques, because it is independent of the type of waveform/distribution. This independence arises from the fact that CA is based on the Central Limit Theorem. The E-MACSC approach provides several benefits as follows: (a) it maintains the prescribed hit ratio efficaciously, (b) it lessens cache size oscillation, and (c) it uses a fixed number of data samples and this makes its computation time more predictable. The E-MACSC is unique because of the following reasons: (a) it utilizes the relative popularity of the data objects as the sole control parameter and (b) it tunes the cache size adaptively by direct data measurement with the CA support. The relative popularity profile of data objects is called popularity distribution (PD) in the E-MACSC context. Any change in the PD's standard deviation indicates a shift of user preference for particular data objects. Monitoring and leveraging this change is the basis for E-MACSC to find a meaningful popularity ratio for deciding how the cache size should be tuned in a dynamic manner.