Legate I apologise for the lack of clarity in my post. The points were supposed to be seperate.
1. Yes Gollum entered from the east according to that quote. Even I manage to understand that.
2. When he first sought the 'thief' having emerged from the Misty Mountains I was saying it was unlikely that on his
"turn towards Mordor" he would enter from the West ( by saying this I am suggesting that in his search he sought west of the Misty Mountains when he came out) and so pass through Moria to cross the Misty Mountains and east toward Mordor.
The fact that this was unlikely I said was because he would not know it. And as you say would not want to pass through it. Thus he used his own 'patch' through the Misty Mountains up by the high pass in order to cross them to journey east.