You say that it could have "easily fallen off" but that's exactly what it did. As you can see from the passage below, it fell off at the worst possible moment, nearly resulting in his capture by the goblins. Unfortunately (for the Ring) it fell back into Bilbo's pocket rather than onto the floor:
Bilbo blinked, and then suddenly he saw the goblins: goblins in full
armour with drawn swords sitting just inside the door, and watching it
with wide eyes, and watching the passage that led to it. They were
aroused, alert, ready for anything.
They saw him sooner than he saw them. Yes, they saw him. Whether it
was an accident, or a last trick of the ring before it took a new
master, it was not on his finger. With yells of delight the goblins
rushed upon him.
A pang of fear and loss, like an echo of Gollum's misery, smote Bilbo,
and forgetting even to draw his sword he struck his hands into his
pockets. And- there was the ring still, in his left pocket, and it
slipped on his finger. The goblins stopped short. They could not see a
sign of him. He had vanished. They yelled twice as loud as before, but
not so delightedly.The Hobbit - Riddles in the Dark
In short, it was trying to escape from him the whole time he had it. Gandalf notes that it would periodically shrink or expand (as well as changing weight) in an attempt to fall off of his finger.
'Bilbo knew no more than he told you, I am sure,' said Gandalf. 'He
would certainly never have passed on to you anything that he thought
would be a danger, even though I promised to look after you. He
thought the ring was very beautiful, and very useful at need; and if
anything was wrong or queer it was himself. He said it was "growing on
his mind". and he was always worrying about it; but he did not suspect
that the ring itself was to blame. Though he had found out that the
thing needed looking after; it did not seem always of the same size or
weight; it shrank or expanded in an odd way, and might suddenly slip
off a finger where it once had been tight.' FOTR - The Shadow of the Past