Sequentially get the outer contour of an Edge Image
You can do the following in order to get a connected contour. Get a few points on the contour say like 100 points or so. And then fit a cubic-spline on it. 

I had done something which might be helpful to you. Below is what I had done.
The project was clustering of cars by shapes. So I had images of cars and I a user had marked the boundaries of the car. Red dots are the manually marked points. Then I had fit a closed cubic splines on these few points. The blue contour is the cubic spline. I then used this contour for my purpose 

I am sure you can do something similar to get the contours. Please be aware that it is possible to get an open curve with cubic splines. Let me also tell you constructing splines from these set of co-ordinate points is simple. What you basically want to do is fit a cubic curve between each set of points. 

Please ask if you need further clarification on what I think.