Have you looked at HMM-based classification tools?

• InterPro Scan
• NCBI CDD

edit: You can also do a BLAST search with sequences on the CATH database here that gives you the most similar fold.

If you could live with SCOP, you could use fastSCOP: http://fastscop.life.nctu.edu.tw/intro.php

From the website: "fastSCOP is a web server that rapidly identifies the structural domains and determines the evolutionary superfamilies of a query protein structure. This server uses 3D-BLAST to scan quickly a large structural classification database (SCOP1.71 with <95% identity with each other) and the top ten hit domains, which have different superfamily classifications, are obtained from the hit lists. MAMMOTH, a detailed structural alignment tool, is adopted to align these top ten structures to refine domain boundaries and to identify evolutionary superfamilies."

"I want to predict folds for the remaining 20% of the structures also."

IMHO, you cannot assign an existing SCOP class to these entries, because these 20% are novel .

It is true that approximately 20% of proteins in PDB is not associated with any known SCOP classification. SCOP curators are well aware of these group of structures. This 20% structures are not assigned a classification due to the sequence-structure diversity of those structures with rest of the members in PDB. For example the structures my contain additional embellishments in structure that will disqualify them from being a member of a particular fold/class. Some of these may get merged into existing classes in the future or depending upon the number of new members with similar features, it may add to a novel fold. As large part of SCOP is manually curate, you may need to wait until SCOP curators assign a fold details to these entries.