Mini-extreme-mass-ratio inspirals (mini-EMRIs), composed of a stellar-mass compact object and a much lighter companion, are promising sources of continuous gravitational waves in the frequency band of ground-based interferometers such as LIGO-Virgo-KAGRA. Such systems, consisting of sub-solar-mass compact objects, provide a unique probe of exotic compact objects, including primordial black holes. Detecting such long-lived signals, however, remains challenging. Here, we adapt standard methods used in searches for quasi-monochromatic signals to search for mini-EMRIs, and derive a statistical framework that explicitly handles spectral leakage. In particular, we introduce a new method that sums along the tracks in the time-frequency plane carved out by possible mini-EMRI signals, which we call ΣTrack. This refinement establishes a general basis for analyzing long-duration transient signals with rapid frequency evolutions, regardless of the underlying mechanism for gravitational-wave emission. We also compute a new semi-analytic sensitivity estimate within our new statistical framework, which is valid under the assumption that the signal is weak with respect to the noise level. We then establish a statistic that quantifies how to discretize the search parameter space for our method, which works for mini-EMRIs, as well as arbitrary signal types. Furthermore, we characterize the pixel omission effect in the generalized frequency-Hough transform, which reduces the critical ratio by ~22% for representative search configurations. We demonstrate that the ΣTrack method eliminates this degradation by selecting and accumulating exactly one pixel per time segment, albeit with an increased computational overhead. Our results provide a foundation for mini-EMRI searches and demonstrate the potential of current ground-based detectors to probe the existence of sub-solar-mass compact objects.