Recent results from IceCube regarding TXS 0506+056 suggest that it may be useful to test the hypothesis of multiple neutrino flares, where each flare is not necessarily accompanied by a corresponding gamma-ray flare. An untriggered, time-dependent, source-stacking search would be optimal for testing this hypothesis, however such an analysis has yet to be applied to the full duration of IceCube data. Here, we discuss one possible way of constructing such an analysis, with a global test statistic obtained by summing the individual test statistics associated with each signal-like flare in the sample. This has the additional advantage of assessing the time structure of each source when running the analysis over a source catalog. We show that, for a signal consisting of many small flares, this style of analysis represents a significant increase in discovery potential over both the existing single-flare fit and the time-integrated stacking methods. Potential source catalogs are examined in combination with this method, including the possibility of a ``self-triggered'' catalog consisting of the locations of the highest energy northern sky events in the IceCube sample.