Infinite-order rogue waves that are small (but not small in $L^2$)

General rogue waves of infinite order constitute a family of solutions of the focusing nonlinear Schr\"odinger equation that have recently been identified in a variety of asymptotic limits such as high-order iteration of B\"acklund transformations and semiclassical focusing of pulses with specific amplitude profiles. These solutions have compelling properties such as finite $L^2$-norm contrasted with anomalously slow temporal decay in the absence of coherent structures. In this paper we investigate the asymptotic behavior of general rogue waves of infinite order in a parametric limit in which the solution becomes small uniformly on compact sets while the $L^2$-norm remains fixed. We show that the solution is primarily concentrated on one side of a specific curve in logarithmically rescaled space-time coordinates, and we obtain the leading-order asymptotic behavior of the solution in this region in terms of elliptic functions as well as near the boundary curve in terms of modulated solitons. The asymptotic formula captures the fixed $L^2$-norm even as the solution becomes uniformly small.