Yes, the SSBN gives away it’s position soon after the first bird has flown. Why? Even if your SSBN has done one of its primary missions, remain undetected and make sure there are no opposing forces nearby, you have just launched a rocket that will have powered flight well into a suborbital trajectory. Even if there is significant cloud cover, once the missile breaks through the clouds, it is visible to every surveillance satellite in the hemisphere. The infrared signature from its rocket motor provides a trail that points right back to its point of origin. This satellite surveillance capability to detect and track ballistic missile launch events is one that every significant nuclear power has, as a part of their Command and Control structure.