Some insights:
Capt McGonagle to NCOI:
LTJG PAINTER came onto the bridge after general
quarters to assume the watch as the officer of the deck. As he assumed
the officer of the deck watch, he indicated that he was having
difficulty in obtaining an accurate ship's fix. At that time, and the
time was approximately 1400, I personally sighted the Minaret at El Arish
to be on a bearing of 142 from the ship and the range as I recall from
the radar was approximately 25.5 miles. I do not know that this radar
range can be verified from the records available at this time.
Confirmation that the bearing log from 12:00 on is not "available."
Ensign David Lucas, who was apparently in charge of surface lookouts, told the NCOI between 12:00 and 1300, “more smoke was visible and the land itself was barely visible. This was through the telescope or high powered binoculars on the 04 level.”
That's just about 20 nmi. How about ships? How far away? Lucas on lookouts:
"They had received refresher training, this was sometime since we left Rota on.the 2nd of June, and had been refreshed on reporting all surface contacts to include relative bearing, approximate range, and target angle. Had also been given refreshers on reporting aircraft properly, on elevation, what angle, other factors pertaining to properly reporting all surface and air contacts to the officer of the deck. This had been stressed heavily in the first week in June, before we did reach our operating area.”
Officer of the deck's notes show no reports of surface craft on June 8 prior to the attack. The main deck log does mention two ships sighted June 7, in this same vigilant mode:
1420 sighted unidentified merchant ship bearing 198, distance about 4800 yards. Identified asmerchant ship as Bencleugh, registry unknown. […] 1430 sighted merchant ship Ioanis Asptiotis Greek registry, bearing 019, distance about 1500 yards.
These were about 2.37nm and 0.74nm distant. There were almost certainly other ships around further out, but these were apparently too far away to notice or care about. So lack of visual contact is not much of a problem.
But, their radar had a range of up to 24.5 miles anyway, judging by ranges given in the chart. This was for a large object called "nearest land (N.L.), not a piddling warship. I hear the Liberty did pick up the MTBs at a range of about 16-18 mi, have to dig that back up. A ship of any size then would have to be probably 20 mi plus away, to not show on radar.
So it's a little ambiguous, but not looking too good for the second mystery ship hypothesis, leaving one wondering where prolonged shelling reports and two ships approaching reports came from in the minutes just after Liberty was erased from the IDF combat info system. The strange radar reading grossly off in all other ways is also left mysterious without another ship, and being in about the right location for Liberty another ship would be ruled out anyway unless under Romulan cloaking shield.
So, the apparent second ship is actually comprised of nothing but a floating ball of random errors, but with set properties, location, speed, direction, and quite uncanny timing! The ball lightning of the earth's coincidence field! The Liberty, like other such politically potent placements of naval units, as always, was the lightning rod.