September, October, and most of November are notorious months for having difficult/bad weather in the Caribbean, all over not just Jamaica, it is Hurricane Season which lasts June to November. It is the end or rather the height of the rainy season or hurricane season in October.
My experience has been that most bad/hurricane/rainy weather happens Sept - Nov. It will still be nice then as it is always warm and sunny in Jamaica, but be warned, it will rain and it will rain this time of year more than mosts. Do a search on google to check it out and you will see the average rainfall for that time of year will be higher then other months. You should be fine, but it is not the best month to go due to the rainy season...I avoid Sept and Oct for that very reason. I have been many times at other times during the year because of this. Again, this is all over the Caribean including the islands in the south and Mexico/Gulf of Mexico. The waters have been warmed over the summer months and this results in more active weather all over the Caribbean.
If you do go, you will be fine. It will rain some but it is nice when it does. Things cool down,not much tho, it gets mellow, nice time for an afternoon nap,...its all good. Be sure to include hurricane insurance in your travel insurance coverage just to make sure. This will ensure that if you have to overstay or possibly leave early, you will be covered. Should, and this is a be should, anything happen, the hotel/resort/place you are staying will take care of you. They are used to the weather and changes it brings so they, Jamaicans, are well used to dealing with it and us for that matter. I've been to Mexico and other islands at this time and again, only noticed an increase in rainfall but also more hurricane warnings then other times of the year....never nothing to worry about tho.