Covering a rectangle with six and seven circles

In a recent article Heppes and Melissen have determined the thinnest coverings of a rectangle with up to five equal circles and also for seven circles if the aspect ratio of the rectangle is either between 1 and 1.34457..., or larger than 3.43017.... In this paper we extend these results. For the gap in the seven circles case we present thin coverings that we conjecture to be optimal. For six circles we determine the thinnest possible covering if the aspect ratio is larger than 3.11803 .... Furthermore, for six and seven circles, we give thin coverings for the remaining range of values, thereby extending our previous conjecture for the square.