Hall was charged after an incident at the club's Christmas party on 19 December and must now serve 10 weeks of the sentence, with a further 10 weeks on licence.
A charge of possessing an offensive weapon was dropped prior to the original hearing at the start of February.
A request for bail pending an appeal has been lodged with the Crown Court and will be heard on Friday.
Hall joined the Shrimpers from non-league side Bromley in the summer and has netted four goals for the League Two side this season.
But the 23-year-old now looks set to miss the remainder of the campaign.