Circularly-linked lists are often implemented using arrays which makes them very fast and by their nature do not require dynamic resizing. You just need a quick check on the read and the write indexes to see if they fell off the end and if so, reset it to zero (or one, whatever).
However, they are generally used for things like input buffers, where the data has no real value once read. Contact lists have lasting value and new contacts will overwrite older contacts once the list fills up, which might be ok unless you overwrite your grandmom who is leaving you a bunch of cash in her will.
I do not think that a linked list is the most efficient way to go for a circular buffer (the original question).
The purpose of a circular buffer is speed and an array simply cannot be beaten for speed in the context of a circular buffer. Even if you keep a pointer to your last accessed linked list item, an array will still be more efficient. Lists have dynamic resizing capabilities (overhead) that are unneeded for circular buffers.
Having said that, I think a circular buffer is probably not the right structure for the application (contact list) you mention.