No other way ?

[The Might]

I just had this sudden idea as I was re-reading LOTR today, The Forbidden Pool.
There Faramir was talking to Frodo about the future of his quest and about the way he should choose. Clearly he was not pleased with the choice made by Gollum (Cirith Ungol), and asked Frodo several times if there was no other way. So, I'll be a copy-cat and ask the same:
No other way at all???
I mean, even if the mountains were steep, hobbits are good climbers, and Gollum as well. Wouldn't it have been much, much easier and less dangerous for them to just climb and pass the mountains somewhere in North Ithilien. That way they would have never had the whole Cirith Ungol problem.

Faramir himself knows no other way, because, as he says, all ways that might have been known were forgotten, and no man of Gondor had entered the mountains:

`Where then shall I go? ' said Frodo. `Back to the Black Gate and deliver myself up to the guard? What do you know against this place that makes its name so dreadful? '
`Nothing certain,' said Faramir. 'We of Gondor do not ever pass east of the Road in these days, and none of us younger men has ever done so, nor has any of us set foot upon the Mountains of Shadow. Of them we know only old report and the rumour of bygone days.

However, is there a hint that there might have been another way across the mountains ?