  A knot $\kappa$ in $S^3$ is persistently foliar if, for each non-trivial
boundary slope, there is a co-oriented taut foliation meeting the boundary of
the knot complement transversely in a foliation by curves of that slope. For
rational slopes, these foliations may be capped off by disks to obtain a
co-oriented taut foliation in every manifold obtained by non-trivial Dehn
surgery on that knot. We show that any composite knot with a persistently
foliar summand is persistently foliar and that any nontrivial connected sum of
fibered knots is persistently foliar. As an application, it follows that any
composite knot in which each of two summands is fibered or at least one summand
is nontorus alternating or Montesinos is persistently foliar.
  We note that, in constructing foliations in the complements of fibered
summands, we build branched surfaces whose complementary regions agree with
those of Gabai's product disk decompositions, except for the one containing the
boundary of the knot complement. It is this boundary region which provides for
persistence.
