In first doing research, I found that most historical coifs had a circular pattern, such as the one pictured below.
http://collections.vam.ac.uk/item/O78811/coif-unknown/
http://collections.vam.ac.uk/item/O251244/coif-and-forehead-unknown/
http://www.metmuseum.org/art/collection/search/228946 (pictured above)
A step by step guide on how to make a coif is available at https://thepragmaticcostumer.wordpress.com/2013/06/05/the-ultimate-one-pattern-piece-project-elizabethan-coif/.
I decided to make my coif with a diamond pattern on it, similar to the coif pictured below.
https://www.pinterest.com/jarnopp/blackwork-embroidery/ (pictured above)
For the floral design, I was inspired by this pattern (pictured below):
This pattern is found at http://www.dragonlore.net/free_patterns.php, which has patterns from a 16th century Italian sampler located in the Victoria & Albert Museum.
Here are pictures to show my progress from start to finish on the coif:
Fun fact: it takes me approximately 20 minutes to stitch one flower.
The strings at the bottom are a purple lucet cord that I made.
