High-assurance systems found in safety-critical infrastructures are facing steadily increasing cyber threats. These critical systems require rigorous guarantees in information flow security to prevent confidential information from leaking to an unclassified domain and the root of trust from being violated by an untrusted party. To enforce bit-tight information flow control, gate-level information flow tracking (GLIFT) has recently been proposed to precisely measure and manage all digital information flows in the underlying hardware, including implicit flows through hardware-specific timing channels. However, existing work in this realm either restricts to two-level security labels or essentially targets two-input primitive gates and several simple multilevel security lattices. This article provides a general way to expand the GLIFT method for multilevel security. Specifically, it formalizes tracking logic for an arbitrary Boolean gate under finite security lattices, presents a precise tracking logic generation method for eliminating false positives in GLIFT logic created in a constructive manner, and illustrates application scenarios of GLIFT for enforcing multilevel information flow security. Experimental results show various trade-offs in precision and performance of GLIFT logic created using different methods. It also reveals the area and performance overheads that should be expected when expanding GLIFT for multilevel security.