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Sequence alignment is common nowadays as it is used in many fields to determine how closely two sequences are related and at times to see how little they differ. In computational biology / Bioinformatics, there are many algorithms developed over the course of time to not only align two sequences quickly but also get good laboratory results from these alignments. The first algorithms developed were based of a technique called Dynamic Programming, which were very slow but were optimal when it comes to sensitivity. To improve speed, more algorithms today are based of heuristic approach, by sacrificing sensitivity. In this paper, we are going to improve on a heuristic algorithm called MASAA (Multiple Anchor Staged Local Sequence Alignment Algorithm) -and MASAA -S which we published previously. This new algorithm appropriately called MMSAA -FG, stands for Maximum Match Subsequence Alignment Algorithm -Finely Grained. The algorithm is based on suffix tree data structure like our previous algorithms, but to improve sensitivity, we employ adaptive seeds, and finely grained perfect match seeds in between the already identified anchors. We tested this algorithm on a randomly generated sequences, and Rosetta dataset where the sequence length ranged up to 500 thousand.