I would say an upgrade looks very promising. Here’s the thing though, even though the socket holding your cpu accepts better cpus …. There is no way to guaranty that your bios/motherboard would support them. Having said that I’ve upgraded loads of laptop cpus and never encountered this problem so I’m optimistic you’d be ok. Don’t sue me if they don’t work though lol. They all have the same wattage (35) as your cpu so overheating should not be an issue and all are Core 2 Duo. For pricing, I’m assuming you are in the U.S.
Your cpu …T6600 … 2.20 ghz … 2 mb L2 cache
Option 1 …T9600 … 2.80 ghz … 6 mb L2 cache … (on ebay for $11)
Option 2 …T9800 … 2.93 ghz … 6 mb L2 cache … (on ebay for $25)
Option 3 …T9900 … 3.06 ghz … 6 mb L2 cache … (on ebay for $48)
I think if it were me, I’d go for option 1. It’s noticeably better than what you have and not too far off the other 2. I would think that at that price it wouldn’t be the end of the world if the cpu turned out to be not supported by your system. Even if you had to put your old processor back in, you would have a spotlessly clean cooling system and fresh thermal paste for your efforts. Are you doing the work yourself? Let me know what you think.