Field-Programmable Gate Arrays (FPGAs) play an indispensable role in Electronic Design Automation (EDA), translating Register-Transfer Level (RTL) designs into gate-level netlists. The correctness and reliability of FPGA logic synthesis tools are critically important, as unnoticed bugs in these tools may infect the final hardware implementations. However, recent approaches often rely heavily on random selection strategies, limiting the structural diversity of the generated HDL test cases and resulting in inadequate exploration of the tool's feature space. To address this limitation, we propose Lin-Hunter, a novel testing framework designed to systematically enhance the diversity of HDL test cases and the efficiency of FPGA logic synthesis tool validation. Specifically, Lin-Hunter introduces a principled set of metamorphic transformation rules to generate functionally equivalent yet structurally diverse HDL test case variants, effectively addressing the limited diversity of existing test inputs. To further enhance bug discovery efficiency, Lin-Hunter integrates an adaptive strategy selection mechanism based on the Linear Upper Confidence Bound (LinUCB) method. This method leverages feedback from synthesis logs of previously executed test cases to dynamically prioritize transformation strategies that have empirically demonstrated a higher likelihood of triggering synthesis bugs. Comprehensive experiments conducted over a three-month period demonstrate the practical effectiveness of Lin-Hunter. Our method has discovered 18 unique bugs, including 10 previously unreported defects, which have been confirmed by official developers. Moreover, our method outperforms state-of-the-art testing methods in both test-case diversity and bug-discovery efficiency.