Here is a slightly different solution. It doesn't necessarily scale well, but it does form a complete cycle of the board.
Solution
First, we create a 5x5 board like this:
+--+--+--+--+--+
| 1|17|24| 2|18|
+--+--+--+--+--+
|22|14| 5|21|13|
+--+--+--+--+--+
|25| 8|11|16| 7|
+--+--+--+--+--+
| 4|20|23| 3|19|
+--+--+--+--+--+
|10|15| 6| 9|12|
+--+--+--+--+--+
Notice that if you move up 3 from the 25th spot, it takes you out of the current 5x5 square. Thus, we can position 4 of these squares to make a 10x10 square, each starting their cycle in the square next to the shared vertex.
Lets call the square above $A$.
Now, define the function $R(S)$ to be the 90 degree counter-clockwise rotation of $S$. Then, we can create a 10x10 square with the following:
R(R(A)) R(A)
R(R(R(A))) A
We'd end up with the following in the middle 6x6 square:
+---+
| 75|
+---+ ...
... | |
+---+---+---+---+
| 51| 26| | 50|
+---+---+---+---+---+---+
|100| | 76| 1 |
+---+---+---+---+
| | ...
... +---+
| 25|
+---+
By doing this, we've created a 10x10 board with a complete cycle.
More General Solution
Consider the following 5x5 squares.
+--+--+--+--+--+
| 1|24| 8| 2|23|
+--+--+--+--+--+
|19|16| 5|20|17|
+--+--+--+--+--+
| 7|10|13|25| 9|
+--+--+--+--+--+
| 4|21|18| 3|22|
+--+--+--+--+--+
|12|15| 6|11|14|
+--+--+--+--+--+
Notice that moving diagonally up and to the right from the 25th spot will take you to the upper left position of the next 5x5 square, and moving diagonally down and to the right will take you to the bottom left position of the next square. Thus, when entering this square from the top left position, you can enter the square to the right in the bottom left or top left position.
For example (S is the starting point in the first square, E is the ending point, and p is the possible starting point in the next square):
+-----+-
|S |p
| |
| E | ...
| |
| |p
+-----+-
A diagonal flip of this square will let you enter the square below in either the top left or top right position when starting in the top left corner.
+-----+
|S |
| |
| |
| E |
| |
+-----+
|p p|
...
Now consider this second square:
+--+--+--+--+--+
| 1|17| 7| 2|14|
+--+--+--+--+--+
| 9|22|25|10|21|
+--+--+--+--+--+
| 6| 3|13|18| 4|
+--+--+--+--+--+
|24|16| 8|23|15|
+--+--+--+--+--+
|12|19| 5|11|20|
+--+--+--+--+--+
Similar to the first, you can enter the square above in either bottom corner. A diagonal flip means you can enter the square to the left in either right corner.
Throught the use of these two squares and their diagonal flip, you are able to enter the corner of any adjacent square when starting in the upper left corner.
Through rotations, this also means you can also start in any corner!
To tile any rectangle which sides are multiples of 5, simply divide it up into 5x5 squares and then trace a path through these squares. A spiral will work, or back and forth through each row will also work. Use the appropriate square above to enter/exit the right corner and it is possible to tile any rectangle whose sides are multiples of 5.
Example
Here is a 30x20 rectangle. This is made up of 6x4 5x5 squares. The starting corner of each 5x5 square is lettered in order.
+--+--+--+--+--+--+
|a |b |c |d |e |f |
| | | | | | |
+--+--+--+--+--+--+
| l| k| j| i| h| g|
| | | | | | |
+--+--+--+--+--+--+
|m |n |o |p |q |r |
| | | | | | |
+--+--+--+--+--+--+
| x| w| v| u| t| s|
| | | | | | |
+--+--+--+--+--+--+
The squares a-e and m-n all start in the top left and enter the square to the right in the top left.
The squares g-k and s-w all start in the top right and enter the square to the left in the top right.
The squares f and r start in the top left and enter the square below in the top right.
The sqaures l and x start in the top right and enter the square below in the top left.
Another way to tile this using a spiral pattern. For example, here is a 7x5 set of 5x5 squares.
+--+--+--+--+--+--+--+
|a |b |c |d |e |f |g |
| | | | | | | |
+--+--+--+--+--+--+--+
| |u |v |w |x |y | h|
|t | | | | | | |
+--+--+--+--+--+--+--+
| | |G |H |I | z| i|
|s |F | | | | | |
+--+--+--+--+--+--+--+
| | | | | | A| j|
|r | E| D| C| B| | |
+--+--+--+--+--+--+--+
| | | | | | | k|
| q| p| o| n| m| l| |
+--+--+--+--+--+--+--+
The pattern is to simply enter the next square in the corner farthest from the center.