score:4
If you were admitted for six months, you can stay for up to six months. You don't need to inform anyone. Even if you want to do so, you will have a hard time finding anyone to accept that information.
You can check your period of admission at https://i94.cbp.dhs.gov.
However, given your earlier visa refusal, you should be careful. It's possible that your change of plans could be taken as a deception designed to help your second visa application succeed. I haven't heard about this happening much in the US, but it is common in the UK. Given the current US political climate, it might be a good idea to be cautious.
If you can show a credible reason for your change of plans, you probably don't need to worry about this, but if you have just decided that you want to stay longer on a whim, it might be safer to leave the US and return.