I remember reading about this somewhere but unfortunately I can't find the source right now. Anyway, if I remember it correctly, since you are on grace period your OPT doesn't get automatically extended meaning you cannot start working for the company until the start date of your H-1. If the H1B petition was filed while you were on OPT, it would have been automatically extended and you would be able to continuing working until a decision on your H-1B application was made. . I don't have a source for this info. So take this info with a grain of salt.

Edit: If it's a cap exempt company then you will be able to start working as soon as your h1 application is approved. For other companies you won't be able to start until Oct 1st