AFAIK the client-side caching feature for Redis is not integrated with the spring-session. Which means that any solution would be more of a workaround.
However the Redis Indexed Web Session is subscribed - by default - to Redis keyspace events, which would mean that whenever the session expires or is deleted you would be notified and could subscribe to the respective event.
See Understanding How Spring Session Cleans Up Expired Sessions for more details.
So a possible solution could be to implement a local cache (perhaps use some 3rd party implementation) and then hook the events to the cache.