Mr Crompton emailed the force's Assistant Chief Constable Andy Holt and head of media Mark Thompson on 8 September, four days before the Hillsborough Independent Panel Report was released.
The county's website indicated that his four-year term as constable -- an elected position that involves, among other duties, serving court documents such as eviction notices and subpoenas to citizens -- was set to expire on December 31, 2014.
Mr Marshall is to take on a new role as chief executive of the National College of Policing next month after more than four years as Hampshire's chief constable.
Chief Constable Sara Thornton has already warned a 16% cut in police budgets over the next four years could lead to "noticeable reductions in service".
"I have to say I'm very proud to be the deputy chief constable - we have a very proud record of service - but in terms of the context of these four officers' behaviour, this is not my proudest day, " she said.