Remember that Frodo started hearing "footpads" within a few hours after entering Moria if I recall correctly, which would have been Gollum following. He had probably been lurking in Moria for some time, possibly coming there with the same Orcs that helped him escape from the Elves of Mirkwood (Just a theory). At any rate, he would have been there perhaps several months at that point. How did he cross the bridge? I suspect his familiarity after those months allowed him to find other ways to escape from Moria besides the East Gate. Possibly a ventilation shaft or other opening. Certainly once he sensed the presence of the ring, he would have found a way to follow it.
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Yet all the while I sit and think of times there were before,
I listen for returning feet and voices at the door.
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