The second E class ship was HMS Enterprise. It was completed in 1926 to the same design as Emerald.
From Wiki:
In the early 1930s, Enterprise was fitted with a prototype twin 6" turret in place of her two forward single mounts; and with the trials proving successful it was retained for the rest of her service career. This turret was later worked into the design of the Leander, Amphion and Arethusa classes. The turret installation occupied less space than the superimposed 'A' and 'B' guns of Emerald, therefore the bridge was placed further forward. The bridge was of a new design, being a single block topped by a director tower, rather than the traditional platforms built around the foremast and wheelhouse topped with a spotting top. This design of bridge would appear in the County-class cruisers.
The early to mid 1930's is also when the ships went from flying off platform to catapult. Whether that work was done at the same time as the upgrade to twin 6" turret, I can not find. The plan I am working off shows both ships as twins from aft through to the fore funnel, then the bridge and gun alterations separates them.
If anybody can help me with information on that point, I will be happy to improve the drawing.