One option is just to use the regex | character to try to match each of the substrings in the words in your Series s (still using str.contains).
You can construct the regex by joining the words in searchfor with |:
>>> searchfor = ['og', 'at']
>>> s[s.str.contains('|'.join(searchfor))]
0 cat
1 hat
2 dog
3 fog
dtype: object
As @AndyHayden noted in the comments below, take care if your substrings have special characters such as $ and ^ which you want to match literally. These characters have specific meanings in the context of regular expressions and will affect the matching.
You can make your list of substrings safer by escaping non-alphanumeric characters with re.escape:
>>> import re
>>> matches = ['$money', 'x^y']
>>> safe_matches = [re.escape(m) for m in matches]
>>> safe_matches
['\\$money', 'x\\^y']
The strings with in this new list will match each character literally when used with str.contains.