diff --git a/seq_smith/_seq_smith.pyi b/seq_smith/_seq_smith.pyi
index 495b726..2e78a35 100644
--- a/seq_smith/_seq_smith.pyi
+++ b/seq_smith/_seq_smith.pyi
@@ -18,6 +18,13 @@ class Alignment:
 
 @final
 class AlignmentFragment:
+    def __init__(
+        self,
+        fragment_type: FragmentType,
+        sa_start: int,
+        sb_start: int,
+        length: int,
+    ) -> None: ...
     @property
     def fragment_type(self) -> FragmentType: ...
     @fragment_type.setter
diff --git a/src/lib.rs b/src/lib.rs
index 295a8f7..a1cc943 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -676,9 +676,10 @@ fn _local_global_align_core(params: AlignmentParams) -> PyResult<Alignment> {
     let mut max_col = 0;
 
     for row in 0..params.sb_len {
-        data.prev_score[row] = std::i32::MIN;
+        let score = params.gap_cost(row as i32 + 1);
+        data.prev_score[row] = score;
         data.hgap_pos[row] = -1;
-        data.hgap_score[row] = std::i32::MIN;
+        data.hgap_score[row] = score.saturating_add(params.gap_open);
     }
 
     for col in 0..params.sa_len {
diff --git a/tests/test_seq_smith.py b/tests/test_seq_smith.py
index 1e26e0d..b693855 100644
--- a/tests/test_seq_smith.py
+++ b/tests/test_seq_smith.py
@@ -3,6 +3,8 @@
 from conftest import AlignmentData
 
 from seq_smith import (
+    AlignmentFragment,
+    FragmentType,
     encode,
     format_alignment_ascii,
     generate_cigar,
@@ -454,3 +456,23 @@ def test_generate_cigar_with_deletion(common_data: AlignmentData) -> None:
     alignment = global_align(seqa, seqb, common_data.score_matrix, common_data.gap_open, common_data.gap_extend)
     cigar = generate_cigar(alignment)
     assert cigar == "1M1D"  # AC vs A-, so A matches, then C is a deletion in seqB (query)
+
+
+def test_local_global_align_overhangs() -> None:
+    # Sequence B (Global) has overhanging tails
+    # sequence A: CCCC
+    # sequence B: AAACCCCAAA
+    # Expected: A aligns to central C's, B has leading/trailing gaps.
+    seqa = encode("CCCC", "ACGT")
+    seqb = encode("AAACCCCAAA", "ACGT")
+    sm = make_score_matrix("ACGT", match_score=2, mismatch_score=-2)
+    aln = local_global_align(seqa, seqb, sm, gap_open=-3, gap_extend=-1)
+
+    expected_fragments = [
+        AlignmentFragment(FragmentType.AGap, 0, 0, 3),
+        AlignmentFragment(FragmentType.Match, 0, 3, 4),
+        AlignmentFragment(FragmentType.AGap, 4, 7, 3),
+    ]
+    assert aln.fragments == expected_fragments
+
+    assert aln.score == -2
diff --git a/uv.lock b/uv.lock
index 1cc9c84..0828b28 100644
--- a/uv.lock
+++ b/uv.lock
@@ -1019,7 +1019,7 @@ wheels = [
 
 [[package]]
 name = "seq-smith"
-version = "0.3.0"
+version = "0.4.0"
 source = { editable = "." }
 dependencies = [
     { name = "numpy", version = "2.2.6", source = { registry = "https://pypi.org/simple" }, marker = "python_full_version < '3.11'" },