Start small
Rain and cold affect dogs differently depending on coat, size, health, and age.
There is no need to rush. A predictable, manageable step is usually more useful than trying to solve the whole situation in one day.
Try this next
Keep outings shorter if needed, dry your dog after the walk, and watch for signs they want to head home.
Notice what helps your dog stay comfortable and engaged, then repeat that version before making the situation harder.
Keep it kind and practical
Keep the next step easy enough that your dog can succeed. If you are worried about safety or feel stuck, pause and ask a qualified reward-based professional for help.