While the plot for the upcoming movie has not been confirmed by Serkis, it seems most likely that it will cover the hunt for Gollum described in the appendices to The Lord of the Rings.

Lord of the Rings: The Hunt for Gollum's makers have delved into why they're bringing back the ring-obsessed ... of his journey we didn't have time to cover in the earlier films.

The original cast of The Lord Of The Rings movies ... It was so different from Rings as an experience. It was a beautiful time, but this idea of kind of going back is like… ‘Absolutely ...