You are given a string, num, which represents a large integer. Your task is to find the largest odd-valued integer that can be formed as a non-empty substring of num. Return this odd integer as a string. If no odd integer exists, return an empty string ("").

Note: A substring is a continuous sequence of characters within a string.

Constraints:

• $1 \leq$ num.length $\leq 10^4$ ...