The 242nd Infantry Regiment was a part of the 42nd Infantry Division, not the 66th Infantry Division.
Neither of these divisions participated in the D-Day invasion of France, 6 June 1944.
The 66th ID left NY P/E, 1 Dec 44; arrived England, 12 Dec 44; arrived France-ETO, 25 Dec 44; entered combat, 1 Jan 45; on line for 91 days; returned to NY P/E, 5 Nov 45; inactivated at Camp Kilmer, NJ, 8 Nov 45.
The 42nd ID left NY P/E, 25 Nov 44; arrived France-ETO, 8 Dec 44; entered combat, 17 Feb 45; on line 106 days; inactivated in Austria, 30 Jun 46.
Which is NOT to say that he did not engage the Germans on Omaha Beach, as you indicate.
It simply suggests that he was probably with some other of the invading forces.
Do you have access to his separation papers ?
They contain some valuable information, including his awards/medals entitlements, and most important in this case, the date of departure from the states, and the date of his return.