There could be spirits in the Timeless Halls more powerful than the Valar
Not all of the Valar, or similarly powerful spirits, descended to Arda initially. Tulkas and Ungoliant entered the world after it was created (though of course before it was fully formed).
It is told among the wise that the First War began before Arda was
full-shaped, and ere yet there was any thing that grew or walked upon
earth; and for long Melkor had the upper hand. But in the midst of the
war a spirit of great strength and hardihood came to the aid of the
Valar, hearing in the far heaven that there was battle in the Little
Kingdom; and Arda was filled with the sound of his laughter. So came
Tulkas the Strong, whose anger passes like a mighty wind, scattering
cloud and darkness before it; and Melkor fled before his wrath and his
laughter, and forsook Arda, and there was peace for a long age.
—The Silmarillion, Quenta Silmarillion: Of the Beginning of Days
Tulkas was among the strongest of the Valar, but he did not intially descend to Arda. Certainly he was stronger than Oromë.
Oromë is a mighty lord. If he is less strong than Tulkas, he is more
dreadful in anger; whereas Tulkas laughs ever, in sport or in war, and
even in the face of Melkor he laughed in battles before the Elves were
born.
—The Silmarillion, Valaquenta
There is also Ungoliant:
The Eldar knew not whence she came; but some have said that in ages
long before she descended from the darkness that lies about Arda, when
Melkor first looked down in envy upon the Kingdom of Manwë, and that
in the beginning she was one of those that he corrupted to his
service. But she had disowned her Master, desiring to be mistress of
her own lust, taking all things to herself to feed her emptiness; and
she fled to the south, escaping the assaults of the Valar and the
hunters of Oromë, for their vigilance had ever been to the north, and
the south was long unheeded.
—The Silmarillion, Quenta Silmarillion: Of the Darkening of Valinor
As detailed in other questions, Ungoliant may have been an Ainu corrupted by Morgoth, either before or after entering into Arda.
Ungoliant's Unlight was capable of confounding the Valar, and when Morgoth turned on her, she attacked him so viciously that he had to call upon his Balrogs for aid.
But Ungoliant had grown great, and he less by the power that had gone
out of him; and she rose against him, and her cloud closed about him,
and she enmeshed him in a web of clinging thongs to strangle him. Then
Morgoth sent forth a terrible cry, that echoed in the mountains.
Therefore that region was called Lammoth; for the echoes of his voice
dwelt there ever after, so that any who cried aloud in that land awoke
them, and all the waste between the hills and the sea was filled with
a clamour as of voices in anguish. The cry of Morgoth in that hour was
the greatest and most dreadful that was ever heard in the northern
world; the mountains shook, and the earth trembled, and rocks were
riven asunder. Deep in forgotten places that cry was heard. Far
beneath the rained halls of Angband, in vaults to which the Valar in
the haste of their assault had not descended, Balrogs lurked still,
awaiting ever the return of their Lord; and now swiftly they arose,
and passing over Hithlum they came to Lammoth as a tempest of fire.
With their whips of flame they smote asunder the webs of Ungoliant,
and she quailed, and turned to flight, belching black vapours to cover
her; and fleeing from the north she went down into Beleriand, and
dwelt beneath Ered Gorgoroth, in that dark valley that was after
called Nan Dungortheb, the Valley of Dreadful Death, because of the
horror that she bred there.
—The Silmarillion, Quenta Silmarillion: Of the Flight of the Noldor
Something to note here is that even if a spirit were not initially as powerful as the Valar, the power of Morgoth (and perhaps the Valar) was lessened from long abiding in Eä, and investing his essence in it: But Ungoliant had grown great, and he less by the power that had gone out of him.
A spirit newly entered, though perhaps only as powerful as a Maia, might still have had the advantage against a spirit of greater innate potency, if the later had dispersed their power. This is basically what happened with Ungoliant, for of a certainty she could not have been created greater than Melkor, who was first in power among all the Ainur.
Despite its incomplete state (whether due to the loss of the
conclusion of the fully developed form of the essay or to its
abandonment, see note 6) this is the most comprehensive account that
my father wrote of how, in his later years, he had come to 'interpret'
the nature of Evil in his mythology; never elsewhere did he write any
such exposition of the nature of Morgoth, of his decline, and of his
corruption of Arda, nor draw out the distinction between Morgoth and
Sauron: 'the whole of Middle-earth was Morgoth's Ring'.
—History of Middle-earth: Morgoth's Ring
This question indicates that when Sauron did not wear his Ring, his power was not diminished.
Morgoth certainly was, though. Perhaps as so much of his power was invested in the creatures of Arda, the result for him was similar to what would have occurred had someone else properly claimed the Ring: that portion of his power was lost.
Note also that a newly arrived destructive spirit might be more powerful than a peaceable one that had been long in Eä.
The Valar 'fade' and become more impotent, precisely in proportion as
the shape and constitution of things becomes more defined and settled.
The longer the Past, the more nearly defined the Future, and the less
room for important change (untrammelled action, on a physical plane,
that is not destructive in purpose). The Past, once 'achieved', has
become part of the 'Music in being'. Only Eru may or can alter the
'Music'.
—History of Middle-earth: Morgoth's Ring
So while the Valar may have been very constrained in later ages, a spirit with the desire only to destroy, i.e. an evil spirit, might not have been.
Entering into Eä, and choosing to stay, essentially makes a spirit a Vala (or Maia)
Tulkas remained and became one of the Valar of the Kingdom of Arda;
but Melkor brooded in the outer darkness, and his hate was given to
Tulkas for ever after.
—The Silmarillion, Quenta Silmarillion: Of the Beginning of Days
It would seem that a connection is drawn between Tulkas's choice to remain, and his becoming one of the Valar. He wasn't initially one of the Valar (despite the apparent symmetry of seven female and seven male Valar). It was his decision to stay in Eä (and of course not to rebel against Eru), that made him such.
It is, however, possible that an Ainu that did not care for Eä would not be bound to it, as the questioner's quote implies:
But this condition Ilúvatar made, or it is the necessity of their
love, that their power should thenceforward be contained and bounded
in the World, to be within it for ever, until it is complete, so that
they are its life and it is theirs.
Indeed, Morgoth may have been able to leave Eä after it was created, and after the Valar had descended, as indicated: "...and Melkor fled before his wrath and his laughter, and forsook Arda, and there was peace for a long age."
It therefore seems very likely that an Ainu that did not love Eä would not be bound to it, unless they had dissipated their power therein. Of course, it is possible that Morgoth did not flee from Eä, but merely from Arda.
Note: I couldn't find any evidence of a name for those Ainur who stayed in the Timeless Halls, besides of course Ainur.
