  We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge
type at primes of good reduction. We make use of the global Serre-Tate
coordinates of Chai as well as recent results of D'Addezio about the $p$-adic
monodromy of isocrystals. The new ingredients in this paper are a general
monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge
type, and a rigidity result for the formal completions of ordinary Hecke
orbits. Along the way we show that classical Serre--Tate coordinates can be
described using unipotent formal groups, generalising results of Howe.
