Callan introduced stabilized-interval-free (SIF) permutations, and Ardila, Rincón, and Williams showed that connected positroids bijectively correspond to SIF permutations unless the underlying set is a singleton. In this paper, we derive an approximate formula for the number of SIF permutations on the set {1,..., N} that refines the previous result conjectured by Callan and proved by Salvatore and Tauraso.