You have asked a great question.  And I am sure there will be many opinions on this topic.  

 

I wonder if you noticed any corrosion at the base of the tower where it meets the ground.  It is not uncommon to find older homes with tv towers buried directly into the ground. My limited experience finds these installations corrode at ground level to the point that you can push your finger through the tower legs.  I know...pretty scary.

 

I think you have two major considerations.  First, what will lighting do if it hits your j-pole antenna?  Probably travel down the coax?

Consider adding a coax ground system at the base of the tower.  

You may want to add polyphaser lightning protector as well.  I've used this company to help me with grounding boxes and I like their work -  https://www.kf7p.com/KF7P/Products.html.   But DX Engineering, Ham Radio Outlet and many others carry these types of products.

 

The next consideration might be to encourage the lighting to spread out as much as possible.  I have three 8' ground rods around my tower.  Each ground rod is bonded to one of the legs of the tower.  Additionally, the three ground rods are bonded together to form a circle around the tower.  I also made additional 10' runs from the ground rods surrounding the tower out into the yard -- something like a star shape.  And the entire system is bonded to the main electrical ground for the utility system. 

 

How much is enough - I don't know.  Do you need any of this?  I have more metal and wire in the air than just a j-pole. 
I attended Ward Silver's talk in Orlando at the recent HamCation.  He's smart.  If you can, get hold of his ARRL book - Grounding and Bonding for the Radio Amateur

Good luck.