In this work we consider a combined theoretical framework comprised by $F(R)$
gravity and a kinetic scalar field. The kinetic energy of the scalar field
dominates over its potential for all cosmic times, and the kinetic scalar
potential is chosen to be small and non-trivial. In this case, we show that the
primordial gravitational wave energy spectrum of vacuum $F(R)$ gravity is
significantly enhanced and can be detectable in future interferometers. The
kinetic scalar thus affects significantly the inflationary era, since it
extends its duration, but also has an overall amplifying effect on the energy
spectrum of pure $F(R)$ gravity primordial gravitational waves. The form of the
signal is characteristic for all these theories, since it is basically flat and
should be detectable from all future gravitational wave experiments for a wide
range of frequencies, unless some unknown damping factor occurs due to some
unknown physical process.