The best time to shoot the supermoon is when you can really show off it's size. Shot solo, up alone in the sky, a supermoon doesn't look any different than any other moon. It lacks any dramatic comparisons to other objects of well-known size. You want to shoot a supermoon when it is lower in the atmosphere, and in proximity to foreground object, with the right focal length, to make it look "super".

That ultimately means that the best time to photograph a supermoon is when it is rising. When it is half-way up the horizon is probably the ideal time to dramatically demonstrate how big a supermoon is. A supermoon hanging over trees, a lone telephone pole, in relation to some well-known mountains, metropolitan city buildings, etc. are also ways to accentuate the size of a supermoon. Moonsets are also plausible, however they often happen extremely late at night, or in certain areas after the sun has risen. You might be able to get some unique shots with a daytime supermoon, but moonrises are generally more dramatic.

Fundamentally, the relative size of the moon in a photograph has a lot more to do with how you photograph it than its proximity to hearth. The moon gets fractionally larger to our eyes (relative to any other moon, as seen when looking at it when it is directly overhead), but not significantly large enough to naturally see any real difference. To really accentuate the size of a supermoon, you also want to use at least a 50mm focal length, longer if possible. Depth of field compression is really what makes a moon get bigger, in the frame, relative to everything else. Shoot a supermoon at 14-17mm, and it will look tiny. Shoot it at 24-50mm, and it will look normal to larger. Shoot it at focal lengths over 50mm, and you can really make it BIG. I've often shot the moon at 100mm to 400mm.

It should be noted that the longer focal length you use, the farther away from any relative "foreground" subjects you will want to be. If you have a nice solo tree you want in the frame while a REALLY HUGE superperigee moon is rising, you will want to make sure you know how far away you need to be at 200mm or 400mm to compress your field, enlarge the moon relative to that tree, and still keep the tree, and any other foreground scenery, in the frame. Trees on moderately distant hills, buildings off a moderate distance, etc. are going to be much more useful for making a good "supermoon" photo than subjects that are much closer and require a wider field of view.

Regarding exposure settings, rather than repeat myself, I'll direct you to my existing answer on that topic: How do I set the proper exposure for nighttime moon photos? There are some basic starting points, however the brightness of the moon depends on a number of factors. Additionally, depending on your exact camera, my general recommendations are to "push" your exposure as far as you can without clipping (i.e. no highlight blinkies when previewing in camera, but exposed as much as possible otherwise.) Shoot in RAW, an with a bright exposure you will have a LOT of control over detail, contrast, and depth in post.