  We extend results on backstepping hybrid feedbacks by exploiting synergistic
Lyapunov function and feedback (SLFF) pairs in a generalized form. Compared to
existing results, we delineate SLFF pairs that are ready-made and do not
require extra dynamic variables for backstepping. From an (weak) SLFF pair for
an affine control system, we construct an SLFF pair for an extended system
where the control input is produced through an integrator. The resulting hybrid
feedback asymptotically stabilizes the extended system when the synergy gap for
the original system is strictly positive. To highlight the versatility of SLFF
pairs, we provide a result on the existence of a SLFF pair whenever a hybrid
feedback stabilizer exists. The results are illustrated on the 3D pendulum.
