Based on reading your job discription you may wish to cover completly the top of the chimney in the next heavy rain fall to best determine if the water is being driven through the top cap.
If you see that the water is stoped then you can confidently hire a Wood Energy Trained Proffessional in your area to install a "Chimney top Damper system" that will not only continue to keep out the unwanted water but it will also help to raise the tempreture inside your flue giving you safer and more economicle use of your wood burning chimney system.
Important: Please do not proceed with this recomendation if your chimney is used for anything other that solid wood fuel burning. i.e. Gas, Oil,etc
P.S. A Chimney top damper system can save you seveal hunderd dollrs annually on your heatting expenses...
Please excuse my grammer and spelling I hope this will either eliminate trial expenditure or hopful give you a sullution