Because Cirdan wasn't particularly looking to get rid of his Ring at all.
By the Third Age the Havens was one of the major remaining Elf strongholds (the others being Rivendell, Lorien, and - to a lesser extent - Mirkwood) and the power of the Three was mostly to preserve, to prevent decay. The Ring may have been actively in use in the Havens and the Elves appear to have suspected where the three Rings were, as we see in The Council of Elrond (and note that Mirkwood is omitted from this list):
What power still remains lies with us, here in Imladris, or with Cirdan at the Havens, or in Lórien.
It's reasonable to surmise that if Gandalf hadn't arrived, or if he had arrived elsewhere, Cirdan would have still been holding his Ring at the end of the Third Age.
The answer to your question is therefore:
- Cirdan didn't give his Ring to Gandalf because he wanted to get rid of it.
- He gave it because he recognized who and what Gandalf was ("for he knew whence he came and whither at last he would return" - Of the Rings of Power and the Third Age).
- Until that point he actually had no intention of giving it away.
- So therefore he would not have had any reason to give it to anyone else either.