As an example, here is the code:
#!/bin/bash
s=00
until [ $s eq 60 ]
do
let "s += 1"
echo $s
done
$s will be 1, 2, 3 etc, but not 01, 02, 03 etc. How can I avoid this deletion of 0? So, I need $s to be 0109, then 10, 11, etc.

Add the zeros when you print it.
Numbers with leading zeros will be treated as octal, so values like 08 and 09 will fail. Therefore it's best to operate on the numbers without the leading zeros and only add the leading zeros when you need to print it. 

