  Gravitational radiation and some global properties of spacetimes can only be
unambiguously measured at future null infinity . This motivates the interest in
reaching it within simulations of coalescing compact objects, whose waveforms
are extracted for gravitational wave modelling purposes. One promising method
to include future null infinity in the numerical domain is the evolution on
hyperboloidal slices: smooth spacelike slices that reach future null infinity.
The main challenge in this approach is the treatment of the compactified
asymptotic region at future null infinity. Evolution on a hyperboloidal slice
of a spacetime including a black hole entails an extra layer of difficulty, in
part due to the finite coordinate distance between the black hole and future
null infinity. Spherical symmetry is considered here as simplest setup still
encompassing the full complication of the treatment along the radial
coordinate. First, the construction of constant-mean-curvature hyperboloidal
trumpet slices for Schwarzschild and Reissner-Nordstr\"om black hole spacetimes
is reviewed from the point of view of the puncture approach. Then, the
framework is set for solving hyperboloidal-adapted hyperbolic gauge conditions
for stationary trumpet initial data, providing solutions for two specific sets
of parameters. Finally, results of testing these initial data in evolution are
presented.
