There are a number of options for playing online chess against local players. The games are played with short time limits so you get plenty of games in a session.
The Ringwood Chess Club runs tournaments on the Lichess platform with events on Monday and Wednesday evenings and Saturday afternoon. For full information on how to register with Lichess and join the tournaments please see the Ringwood website.
The Wimborne Chess Club also runs a tournament on Thursday evening on Lichess. For joining information please see the Wimborne website.
Nationally the main organisers are the ECF and 4NCL.
A number of New Milton members play regularly and highlights are shown on the Chess News page.