diff --git a/s2/decode.go b/s2/decode.go
index 2aba9e272d..b5fa4d3f00 100644
--- a/s2/decode.go
+++ b/s2/decode.go
@@ -791,6 +791,7 @@ func (r *Reader) Skip(n int64) error {
 			} else {
 				// Skip block completely
 				n -= int64(dLen)
+				r.blockStart += int64(dLen)
 				dLen = 0
 			}
 			r.i, r.j = 0, dLen
@@ -921,6 +922,15 @@ func (r *Reader) ReadSeeker(random bool, index []byte) (*ReadSeeker, error) {
 	err = r.index.LoadStream(rs)
 	if err != nil {
 		if err == ErrUnsupported {
+			// If we don't require random seeking, reset input and return.
+			if !random {
+				_, err = rs.Seek(pos, io.SeekStart)
+				if err != nil {
+					return nil, ErrCantSeek{Reason: "resetting stream returned: " + err.Error()}
+				}
+				r.index = nil
+				return &ReadSeeker{Reader: r}, nil
+			}
 			return nil, ErrCantSeek{Reason: "input stream does not contain an index"}
 		}
 		return nil, ErrCantSeek{Reason: "reading index returned: " + err.Error()}
diff --git a/s2/index_test.go b/s2/index_test.go
index 38975df21b..9a43c2be74 100644
--- a/s2/index_test.go
+++ b/s2/index_test.go
@@ -7,6 +7,7 @@ import (
 	"io/ioutil"
 	"math/rand"
 	"sync"
+	"testing"
 
 	"github.com/klauspost/compress/s2"
 )
@@ -99,3 +100,126 @@ func ExampleIndex_Load() {
 	//Successfully skipped forward to 4444440
 	//Successfully skipped forward to 4999995
 }
+
+func TestSeeking(t *testing.T) {
+	compressed := bytes.Buffer{}
+
+	// Use small blocks so there are plenty of them.
+	enc := s2.NewWriter(&compressed, s2.WriterBlockSize(16<<10))
+	var nElems = 1_000_000
+	var testSizes = []int{100, 1_000, 10_000, 20_000, 100_000, 200_000, 400_000}
+	if testing.Short() {
+		nElems = 100_000
+		testSizes = []int{100, 1_000, 10_000, 20_000}
+	}
+	testSizes = append(testSizes, nElems-1)
+	//24 bytes per item plus \n = 25 bytes per record
+	for i := 0; i < nElems; i++ {
+		fmt.Fprintf(enc, "Item %019d\n", i)
+	}
+
+	index, err := enc.CloseIndex()
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	for _, skip := range testSizes {
+		t.Run(fmt.Sprintf("noSeekSkip=%d", skip), func(t *testing.T) {
+			dec := s2.NewReader(io.NopCloser(bytes.NewReader(compressed.Bytes())))
+			seeker, err := dec.ReadSeeker(false, nil)
+			if err != nil {
+				t.Fatal(err)
+			}
+			buf := make([]byte, 25)
+			for rec := 0; rec < nElems; rec += skip {
+				offset := int64(rec * 25)
+				//t.Logf("Reading record %d", rec)
+				_, err := seeker.Seek(offset, io.SeekStart)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				_, err = io.ReadFull(dec, buf)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				expected := fmt.Sprintf("Item %019d\n", rec)
+				if string(buf) != expected {
+					t.Fatalf("Expected %q, got %q", expected, buf)
+				}
+			}
+		})
+		t.Run(fmt.Sprintf("seekSkip=%d", skip), func(t *testing.T) {
+			dec := s2.NewReader(io.ReadSeeker(bytes.NewReader(compressed.Bytes())))
+			seeker, err := dec.ReadSeeker(false, nil)
+			if err != nil {
+				t.Fatal(err)
+			}
+			buf := make([]byte, 25)
+			for rec := 0; rec < nElems; rec += skip {
+				offset := int64(rec * 25)
+				//t.Logf("Reading record %d", rec)
+				_, err := seeker.Seek(offset, io.SeekStart)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				_, err = io.ReadFull(dec, buf)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				expected := fmt.Sprintf("Item %019d\n", rec)
+				if string(buf) != expected {
+					t.Fatalf("Expected %q, got %q", expected, buf)
+				}
+			}
+		})
+		t.Run(fmt.Sprintf("noSeekIndexSkip=%d", skip), func(t *testing.T) {
+			dec := s2.NewReader(io.NopCloser(bytes.NewReader(compressed.Bytes())))
+			seeker, err := dec.ReadSeeker(false, index)
+			if err != nil {
+				t.Fatal(err)
+			}
+			buf := make([]byte, 25)
+			for rec := 0; rec < nElems; rec += skip {
+				offset := int64(rec * 25)
+				//t.Logf("Reading record %d", rec)
+				_, err := seeker.Seek(offset, io.SeekStart)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				_, err = io.ReadFull(dec, buf)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				expected := fmt.Sprintf("Item %019d\n", rec)
+				if string(buf) != expected {
+					t.Fatalf("Expected %q, got %q", expected, buf)
+				}
+			}
+		})
+		t.Run(fmt.Sprintf("seekIndexSkip=%d", skip), func(t *testing.T) {
+			dec := s2.NewReader(io.ReadSeeker(bytes.NewReader(compressed.Bytes())))
+
+			seeker, err := dec.ReadSeeker(false, index)
+			if err != nil {
+				t.Fatal(err)
+			}
+			buf := make([]byte, 25)
+			for rec := 0; rec < nElems; rec += skip {
+				offset := int64(rec * 25)
+				//t.Logf("Reading record %d", rec)
+				_, err := seeker.Seek(offset, io.SeekStart)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				_, err = io.ReadFull(dec, buf)
+				if err != nil {
+					t.Fatalf("Failed to seek: %v", err)
+				}
+				expected := fmt.Sprintf("Item %019d\n", rec)
+				if string(buf) != expected {
+					t.Fatalf("Expected %q, got %q", expected, buf)
+				}
+			}
+		})
+	}
+}