Coloring the Chess Board v2

Question

This is a follow up question to Coloring the Chess Board

We have 8x8 Chess board like grid (8x8 2D matrix) and we are trying to color every single square/cell. But the rules are interesting;

• You may start from any square you want and color that square for the beginning.
• You can continue coloring another square in the grid which is $4$ or $5$ squares away from the square you started and colored from and so on (vertically or horizontally).
• You cannot color the same square you colored before or come back again on that.

This time, you are not allowed to start from any square you want; you need to start from a corner square (in other words, in chess language a1,a8,h1,h8).

so

What is the maximum amount of squares you can color with the given rule above now?

Answer 1

We can get

64

Like this:

Fill everything except file e and ranks 4-5.
a1 a6 a2 a7 a3 a8
f8 f3 f7 f2 f6 f1
b1 b6 b2 b7 b3 b8
g8 g3 g7 g2 g6 g1
c1 c6 c2 c7 c3 c8
h8 h3 h7 h2 h6 h1
d1 d6 d2 d7 d3 d8

Fill rank 4
d4 h4 c4 g4 b4 f4 a4 e4

Fill the e-file
e8 e3 e7 e2 e6 e1 e5

Fill the rest of rank 5
a5 f5 b5 g5 c5 h5 d5