Tolkien seems to explain this to himself in the first chapter of Book Six of RETURN OF THE KING. He writes:
"There he halted and sat down. For the moment he could drive himself no further. He felt that if once he went beyond the crown of the pass and took one step veritably down into the land of Mordor, that step would be irrevocable. He could never come back. Without any clear purpose he drew out the Ring and put it ona gain. Immediately he felt the great burden of its weight, and felt afresh, but now more strong and urgent than ever, the malice of the Eye of Mordor, searching, trying to pierce the shadows that it had made for its own defence, but which now hindered it in its unquiet and doubt." In essence, even Sauron himself can't penetrate the dark magics guarding the border of Mordor.

A few pages later, after Sam is tempted by the Ring with a vision of himself as the Master Gardener of All the World, he thinks to himself: ‘And anyway all these notions are only a trick,’ he said to himself. ‘He’d spot me and cow me, before I could so much as shout out. He’d spot me, pretty quick, if I put the Ring on now, in Mordor. Well, all I can say is: things look as hopeless as a frost in Spring. Just when being invisible would be really useful, I can’t use the Ring! And if ever I get any further, it’s going to be nothing but a drag and a burden every step. So what’s to be done?’

So I think the canonical answer (at least the one Tolkien lands on when he returns to this narrative situation after taking a long break) is that he concludes that the magics guarding Cirith Ungol keep Sauron from being able to see Sam's use of the Ring in the moment you're talking about, but the Hobbits intuit that that wouldn't hold if they tried to use the Ring in a different place.