Based on the data you provide, it seems that hierarchical clustering (see wiki here) with type "agglomerate" (bottom up) solves your problem, i.e.:

out = FindClusters[data, 6, Method -> "Agglomerate"];
ListPointPlot3D[out]

and get:

Based on how your full dataset looks like (e.g. if you know how many clusters there are etc.), you might need to adapt the code a bit (possibly also with distance function or matrix) - but for the sample you are providing, it seems to work nicely.

Based on the data you provide, it seems that hierarchical clustering (see wiki here) with type "agglomerate" (bottom up) solves your problem, i.e.:

out = FindClusters[data, 6, Method -> "Agglomerate"];
ListPointPlot3D[out]

and get:

Based on how your full dataset looks like (e.g. if you know how many clusters there are etc.), you might need to adapt the code a bit (possibly also with distance function or matrix) - but for the sample you are providing, it seems to work nicely. Also, performance could be an issue, as noted in the wiki article.