  In this paper, we present a new sufficiency condition to obtain a graceful
labelling for every $kC_{4n}$-snake and use this condition to label every such
snake for $n=1,2,\ldots,6$. Then, we extend this result to cyclic snakes where
the cycles vary. Also, we obtain new results on the (near) graceful labelling
of cyclic snakes based on cycles of length $n=6, 10, 14$, completely solving
the case for $n=6$.
