Well, first you need to find a logical connection point between the private story and the main plot. Make sure it doesn't feel forced or out of place.
To do this, you could start by identifying common themes or character traits. Maybe a character from the main story has a hidden backstory that ties in with the private one. Another approach is to use a significant event as a bridge to bring the two together smoothly.
I think you should analyze the structure and pacing of the main story. Then, introduce elements of the private story gradually, building up to a climax or resolution that enriches the overall narrative. It takes some careful planning and seamless integration to make it work effectively.