  We give a diagrammatic characterization of the $(1,1)$ knots in the
three-sphere and lens spaces which admit large Dehn surgeries to manifolds with
Heegaard Floer homology of next-to-minimal rank. This is inspired by a
corresponding result for $(1,1)$ knots which admit large Dehn surgeries to
manifolds with Heegaard Floer homology of minimal rank due to
Greene-Lewallen-Vafaee.
