This is going to sound completely crazy but I have done it and won more work and client loyalty.
When I can’t stand a site I built because it has aged. I either redo part of the site (on my test site –sandbox) or work up wireframes and demonstrate them to the client. This I do for free and is speculative. I’ve never had them not WANT the new design. They see that you have their best interest at heart. It becomes an easy sale. You’ll have to evaluate the personalities of the decision-makers and go with your instincts. I have to be prepared for disappointment but rarely have I been turned down when I produce a “free gift.”
This is a psychological principle of influence called reciprocity. When you do something nice for someone, they naturally want to do something nice for you. Use it to your advantage.