The Secret Fire was never within Morgoth's reach
The Flame Imperishable was, always and ever, solely of Eru Ilúvatar:
To Melkor among the Ainur had been given the greatest gifts of power and knowledge, and he had a share in all the gifts of his brethren. He had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own, and it seemed to him that Ilúvatar took no thought for the Void, and he was impatient of its emptiness. Yet he found not the Fire, for it is with Ilúvatar.
—The Silmarillion, p. 3
Tolkien was a linguist and a philologist. When he used words, he used them with great care, often employing deliberate archaisms to enhance the mythological feel of his work.
Thus, when Tolkien says that the Fire is with Ilúvatar, he means it as a universal, timeless fact.
If the fact that the fire was with Ilúvatar prevented Morgoth when Eä was not yet born, it would present the same obstacle once Eä had been fashioned.
Indeed, it seems clear that the Secret Fire, or the Flame Imperishable, is indeed merely another name for the innate creative power of Eru, which is lacking in his subcreation.
This is actually already glimpsed in the Ainulindale', in which reference is made to the 'Flame Imperishable'. This appears to mean the Creative activity of Eru (in some sense distinct from or within Him), by which things could be given a 'real' and independent (though derivative and created) existence. The Flame Imperishable is sent out from Eru, to dwell in the heart of the world, and the world then Is, on the same plane as the Ainur, and they can enter into it. But this is not, of course, the same as the re-entry of Eru to defeat Melkor. It refers rather to the mystery of 'authorship', by which the author, while remaining 'outside' and independent of his work, also 'indwells' in it, on its derivative plane, below that of his own being, as the source and guarantee of its being.
In the early versions of Tolkien's mythos described in Morgoth's Ring, we may see how futile any attempt by Morgoth (then Melkor) to acquire the Secret Fire would actually be:
Now the Sun was designed to be the heart of Arda, and the Valar purposed that it should give light to all that Realm, unceasingly and without wearying or diminution, and that from its light the world should receive health and life and growth. Therefore Varda set there the most ardent and beautiful of all those spirits that had entered with her into Ea, and she was named Ar(i), and Varda gave to her keeping a portion of the gift of Iluvatar so that the Sun should endure and be blessed and give blessing. The Sun, the loremasters tell us, was in that beginning named As (which is as near as it can be interpreted Warmth, to which are joined Light and Solace), and that the spirit therefore was called Azie (or later Arie).
Note the similarity of this passage with the later description of Eä.
Therefore Ilúvatar gave to their vision Being, and set it amid the Void, and the Secret Fire was sent to burn at the heart of the World; and it was called Eä.
—The Silmarillion, p. 7
The "gift of Ilúvatar," provided to the spirit of the Sun, is a direct analogue of the Secret Fire that burns at the heart of Arda. As previously mentioned, this is simply that element of the divine creative principle that gives Arda its continuing existence. Of course, Morgoth desires this principle for himself, and resolves to take it by force. Arie censures him, saying that he must ultimately be unsuccessful in claiming it.
But Arie rejected Melkor and rebuked him, saying: 'Speak not of right, which thou hast long forgotten. Neither for thee nor by thee alone was Ea made; and thou shalt not be King of Arda.
Beware therefore; for there is in the heart of As a light in which thou hast no part, and a fire which will not serve thee. Put not out thy hand to it. For though thy potency may destroy it, it will burn thee and thy brightness will be made dark.'
Then the spirit of Arie went up like a flame of anguish and wrath, and departed for ever from Arda, and the Sun was bereft of the Light of Varda, and was stained by the assault of Melkor. And being for a long while without rule it flamed with excessive heat or grew too cool, so that grievous hurt was done to Arda and the fashioning of the world was marred and delayed, until with long toil the Valar made a new order. But even as Arie foretold, Melkor was burned and his brightness darkened, and he gave no more light, but light pained him exceedingly and he hated it.
Note that Morgoth did not gain dominion over, or corrupt, the Light of Varda, what Tolkien would later call the Flame Imperishable. He did indeed corrupt the sun, but the Light of Varda, which elsewhere is identified with the love of Eru for creation, departs the world with Arie. It is indeed this that burns Morgoth.
To Varda Iluvatar said: 'I will give unto thee a parting gift. Thou shalt take into Ea a light that is holy, coming new from Me, unsullied by the thought and lust of Melkor, and with thee it shall enter into Ea, and be in Ea, but not of Ea.'
Morgoth could destroy the manifestation of the Flame in Arda, much as he could destroy other manifestations of the Flame Imperishable, such as living creatures.
The Light, or the Flame Imperishable, cannot be destroyed, being (as it is) quite literally an attribute of Ilúvatar.
Any attempt to claim the Secret Fire would only have been to Morgoth's regret. While he could taint its manifestations, as he indeed tainted Arda by pouring his being into it, the Fire itself was entirely part of Ilúvatar.