  Hybrid inflation faces two well-known problems: the blue spectrum of the
non-supersymmetric version of the model and the fine-tuning of the initial
conditions of the fields leading to sufficient inflation to account for the
standard cosmological problems. They are investigated by studying the exact
two-fields dynamics instead of assuming slow-roll. When the field values are
restricted to be less than the reduced Planck mass, a non-negligible part of
the initial condition space (around 15% depending on potential parameters)
leads to successful inflation. Most of it is located outside the usual
inflationary valley and organized in continuous patterns instead of being
isolated as previously found. Their existence is explained and their properties
are studied. This shows that no excessive fine-tuning is required for
successful hybrid inflation. Moreover, by extending the initial condition space
to planckian-like or super-planckian values, inflation becomes generically
sufficiently long and can produce a red-tilted scalar power spectrum due to
slow-roll violations. The robustness of these properties is confirmed by
conducting our analysis on three other models of hybrid-type inflation in
various framework: "smooth" and "shifted" inflation in SUSY and SUGRA, and
"radion assisted" gauge inflation. A high percentage of successful inflation
for smooth hybrid inflation (up to 80%) is observed.
