1. Bitaw ng gas, apak ng clutch.
2. Shift to neutral.
3. Bitaw ng clutch, then rev slightly.
4. Apak ng clutch, shift to the new gear.
5. And off you go.
This technique is required if you're driving an older car. Nowadays, we have the luxury of synchronizers, so this is now classified as a forgotten art. In your case, this will help habang hindi mo pa napapagawa yung transmission mo.
But to answer your question, yes the gear oil change might help. If it doesn't baka upod na yung synchronizers mo.