Patching tarpaper on a roof - do sides of patch go over or under?

[Stephen Tashiro]

I need to repair the big hole in a shingled roof that was left when an evaporative cooler was removed. The hole in the wood has been closed with boards. Next I need to patch the hole in the tar paper. How is new tarpaper attached? Presumably, on the upward slope, I put the edge of the new tar paper under the edge of the old tar paper. On the downward slope the edge of the new tar paper goes over the old tar paper. But what happens on the sides?

[Karl Loeblein]

I would just put a little roofing sealant over the side seems (or any tears), and then install the new shingles. Just make sure no two seams line up between layers, or use more sealant as necessary.

[Andrew Hughes]

Karl has it right.
When I first read I missed the shingle part.
So I deleted my first post.
Sorry for any confusion.

[Jason Roehl]

No need for sealants under the roofing felt. You have accurately deduced the build-up for the top and bottom. The sides just need to overlap by at least 6”. Felt is just a secondary barrier anyway for wind-driven rain that makes its way under the shingles.

[Florian Stauss]

That’s how I would go about it.
If you have the edges of the existing tarpaper exposed for a few inches, you cut the patch a few inches bigger than the hole that needs to be patched.
You can then make a couple of cuts on either side of the bottom or top edge, parallel to the roof ridge. So you can slide the patch in, going under the top and on top of the bottom.
You seal the sides with ice and water barrier or some flashing tape.
Be careful when using roofing cement under the shingles, not to create any ridges or humps that keep the shingles from laying flat.
Good luck