  Covariance functions are the core of spatial statistics, stochastic
processes, machine learning as well as many other theoretical and applied
disciplines. The properties of the covariance function at small and large
distances determine the geometric attributes of the associated Gaussian random
field. Having covariance functions that allow to specify both local and global
properties is certainly on demand. This paper provides a method to find new
classes of covariance functions having such properties. We term these models
hybrid as they are obtained as scale mixtures of piecewise covariance kernels
against measures that are also defined as piecewise linear combination of
parametric families of measures. In order to illustrate our methodology, we
provide new families of covariance functions that are proved to be richer with
respect to other well known families that have been proposed by earlier
literature. More precisely, we derive a hybrid Cauchy-Mat\'ern model, which
allows us to index both long memory and mean square differentiability of the
random field, and a hybrid Hole-Effect-Mat\'ern model, which is capable of
attaining negative values (hole effect), while preserving the local attributes
of the traditional Mat\'ern model. Our findings are illustrated through
numerical studies with both simulated and real data.
