The main theorem is that in the moduli of degree 5 surfaces and degree 6 surfaces, there cannot be any 'high-dimensional' Hodge-theoretically special loci with general Picard rank equal to 1. We use this theorem to deduce that there can only be finitely many Picard-exceptional (in the sense of Baldi-Klingler-Ullmo) components in the moduli of degree 5 surfaces, and that there are only finitely many Picard-exceptional or exceptional components in the moduli of degree 6 surfaces