How to merge two tangled convex hulls (like this) to form a Convex Hull using Graham's Scan or anyother algorithm in linear time?
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Basically, you use Andrew's modification of the Graham scan algorithm.
After the points are sorted in xy-order, the upper and lower hulls are computed in linear time. Since you already have the two convex hulls , the xy-sorting of the convex hull points will also take linear time (e.g., reverse the lower hulls and merge-sort four sorted lists).
Since the rest of the algorithm will take linear time, the overall runtime is linear as you requested.
Here is a reference to some short python code implementing Andrew's modification to Graham's scan. See also my answer here.