You’re sure it’s the laptop socket that’s faulty, and not the headphone plug, I presume. With a magnifying glass and the light shining into the socket as best as you can get it, you may be able to see something. There’s a small chance that you can bend a contact using a pin, but you are quite likely to make it worse. The left channel is the tip of the jack plug and the right (giving trouble) is the next ring on the plug. The troublesome contact will therefore be around a quarter of an inch from the bottom of the socket, i.e. half way down.
Most laptops are fairly straightforward to take apart and this seems to be no exception (https://www.youtube.com/watch?v=WrMbmV2daAw). From the video, I can’t see where the headphone socket is but replacing it shouldn’t require more than average soldering skills. You just need to make sure the replacement has the pins and fixing tags in the same places. You might be able to get a spare for your specific laptop on eBay.