To add to what's already been said numerous times. 6 1/2 -> 5G is not an "upgrade" it's just a change in rim size. The backbore and throat are exactly the same, and the rim diameter is 0.1mm wider. Many people play both pieces and one is not "better" than the other, the only reason people think 6.5 is "basic" is because it usually ships with most student level instruments.

99% of the time, different size mouthpieces are for different size mouths or different size instruments. End of story.

By all means, try a 5G, they're one of the most common mouthpieces so you're sure to find someone who can lend you one to try. If it feels better, give it a crack. If it feels worse, leave it.

Just because you see some orchestral players on huge mouthpieces don't be fooled into thinking you need one. They use big mouthpieces and drill out the throat sizes to get HUGE sound to cut through big orchestras in giant symphony halls. When you get to that stage, you should have a fair idea what mouthpiece works best.

Until then, save the cash and do some more practice.