Logically Necessary but NOT Tautologically necessary? Help

I'm trying to come up with a sentence that is Logically Necessary but NOT Tautologically necessary and I am having a difficult time.

Here is an example of what I mean by "logically necessary": "Notice that according to this definition there are some sentences that are logical consequences of any set of premises, even the empty set. This will be true of any sentence whose truth is itself a logical necessity. For example, given our assumptions about fol, the sentence a = a is necessarily true. So of course, no matter what your initial premises may be, it will be impossible for those premises to be true and for a = a to be false—simply because it is impossible for a = a to be false! We will call such logically necessary sentences logical truths."

'Tautological' means 'in virtue of its truth-functional properties'.

$a=a$ is a perfectly good example. It is a logically necessary truth, but it is not true on the basis of any truth-functional operators involved. So, it is not a tautologically necessary truth.

This is why first-order logic (FOL) is more powerful than propositional logic (PL): FOL can 'see' things that PL cannot.

OK, one more example: 'Either everything is purple, or there is something that is not purple'. This is not a tautological truth: again, PL is unable to 'understand' the quantifiers involved here that make this a logically necessary truth.