Thickness of cat6 depends on the gauge, if its shielded, and if there is a center spine. I'm pretty sure cat6 and up all have spines that their 4 pairs wind around at the very minimum. The lower the gauge the thicker the cable. So although 26 or 28 AWG cables are thinner than 24 awg there is more resistance and the cables will start to heat up more. They sell 26,28,30awg cat6 cables btw. For me i wouldnt ever go higher than 24 AWG. Most cat5e cables are typically 24awg. Cat6 shielding (ie, S/UTP, S/FTP, SF/FTP, U/FTP) is likely not necessary for residential use and also will be more hassle to properly ground them.
So just get cat 6 U/UTP 24awg cables. cat 6 unshielded cables naturally have shielding vs cat5e unshielded cable due to the tighter twists and the pairs winding around its spine.
In all variations discussed cat6 will always be slightly thicker than cat5e. Unless you're willing to go to a much higher gauge like 28awg.