The solution will depend on the design and current height of your fence along with what height you hope to reach.

Fence is 5ft high. Hope to reach 8ft. Its the most basic design

The trick will be matching the current brick. A lot of ways to end up looking like an afterthought. You may also need to check code for your area -- you may need a variance for a fence above a certain height, if it's on the property line.

Not ideal but doable.

I like the lattice. You can grow vines which climb up the wall to the lattice. Would give privacy and look nice.

Most municipalities do not allow fences over 6ft. tall. That's why hedges are often placed in front of the fences to get past the height limit.

Check and find out what is allowed in your area. Most places I've lived the restrictions are 8ft, just tall enough to keep a large dog from jumping. My current area doesn't restrict fence heights and the deer can easily clear an 8ft fence.

More masonry, or a steel or wrought iron rail. Do you have a photo and an idea of height to be added?

Perhaps lattice (that suits/compliments), add shrubs and plants to break up brick and vines will eventually visually connect both sections. Eight feet may be difficult though.