client.go  | 19 +++++++++++++++++++
 torrent.go |  3 ++-

diff --git a/client.go b/client.go
index ae246f9fa72759219c08db43e9ce5fb8083d16f4..5170db9c0c855b693bd530a6b860ad47439fa5bc 100644
--- a/client.go
+++ b/client.go
@@ -130,6 +130,7 @@ 	if int(index) >= len(t.Pieces) {
 		err = io.EOF
 		return
 	}
+	t.lastReadPiece = int(index)
 	piece := t.Pieces[index]
 	if !piece.EverHashed {
 		cl.queuePieceCheck(t, index)
@@ -714,7 +715,25 @@
 func (ResponsiveDownloadStrategy) FillRequests(t *torrent, c *connection) {
 	for e := t.Priorities.Front(); e != nil; e = e.Next() {
 		if !c.Request(e.Value.(request)) {
+			return
+		}
+	}
+	for i := t.lastReadPiece; i < t.lastReadPiece+5 && i < t.NumPieces(); i++ {
+		for cs := range t.Pieces[i].PendingChunkSpecs {
+			if !c.Request(request{pp.Integer(i), cs}) {
+				return
+			}
+		}
+	}
+	// Then finish off incomplete pieces in order of bytes remaining.
+	for _, index := range t.piecesByPendingBytes() {
+		if t.PieceNumPendingBytes(index) == t.PieceLength(index) {
 			break
+		}
+		for chunkSpec := range t.Pieces[index].PendingChunkSpecs {
+			if !c.Request(request{index, chunkSpec}) {
+				return
+			}
 		}
 	}
 }
diff --git a/torrent.go b/torrent.go
index 4f94b89bc0bb842e78c2ba907f0eeba09cab2cf8..39d85e5d7185c11f21219c51169775c2a511e37d 100644
--- a/torrent.go
+++ b/torrent.go
@@ -34,7 +34,8 @@ 	Peers      []Peer
 	Priorities *list.List
 	// BEP 12 Multitracker Metadata Extension. The tracker.Client instances
 	// mirror their respective URLs from the announce-list key.
-	Trackers [][]tracker.Client
+	Trackers      [][]tracker.Client
+	lastReadPiece int
 }
 
 func (t *torrent) String() string {