No, a later trial is not allowed
A prosecutor can, and often will charge multiple related crimes, and all will be addressed at the same trial. But once a person has been acquitted on a given set of events, the same jurisdiction cannot re-try the same person on what is often called a lesser included offense. Nor on a greater offense implied by the same events. Not even if additional evidence comes to light.
However, if an act (or set of acts) is a crime under both state law and Federal law, for example theft by deception (state crime) and wire fraud (federal crime) one jurisdiction may try the person even after ther has been an acquittal in the other.
I think the same rule applies if an act is a crime within the jurisdiction of two different states, that both can trey the accused.
In many cases prosecutors will choose not to bring the second trial, but they can if they see fit.