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No, going deeper into the moon would <i>reduce</i> gravity.

This is covered by the <i>shell theorem of gravity</i>, proved by Newton centuries ago.  If you have an evenly-distributed mass shaped as a hollow sphere, then two things follow:<ol>

<li>Outside the shell, the gravity is the same as if the mass was all at the center of mass point.

<li>Inside the shell there is no gravity.  All the gravitational attractions to the various parts of the shell cancel, regardless of where you are inside the hollow shell.

</ol>