2nd, 3rd, Nth closest guesses I have used the KMeans algorithm to create an engine that can guess the cluster that a particular set of input data will fall into.

Can I use it to guess the 2nd closest cluster, 3rd closest, and so on?

Currently, I am using the sklearn.cluster.KMeans library if that's any help - it doesn't seem like the API provides this functionality already.
You could simply compute the distance to each cluster center provided in the cluster_centers_ attribute (once the KMeans instance is fitted). The predict method actually does that for the closest cluster center.