New shoots on autumn-fruiting raspberries

Last year my autumn-fruiting raspberries (variety unknown) produced new, tall, healthy canes around this time, while the older canes were still fruiting.  I put it down to the weather and, when the time came to prune them (a job I do just after Christmas) I cut down the new along with the old.  The same thing is happening this year.  Should I leave the new shoots to continue growing rather than cutting them back.  Maybe I would get summer fruit from them?  Any advice welcomed.

Mine are the same this year.  I've decided to leave one healthy shoot per plant, in the hope that I'll get some earlier fruit next year (I think it's called 'double cropping'), but don't want to detract too much from the Autumn harvest....

Thanks Victoria.  Seems like a good idea so I think I will do the same and see what happens next year.