How natural is a small but nonzero cosmological constant?

Based on our previous attempt, we propose a better way to understand a small but nonzero cosmological constant, as indicated by a number of recent observational studies. We re-examine the assumptions of our model of two scalar fields, trying to explain the basic mechanism resulting in a series of mini-inflations occuring nearly periodically with respect to $\ln t$ with $t$ the cosmic time. We also discuss how likely the solution of this type would be, depending on the choice of the parameters.