As the doctor has already informed you, due to pregnency, your abdominal muscles could have become weak. Hence the umblicus is slightly protuding. Please continue the abdominal muscle strengthening exercises your doctor has adviced. After delivery, you can do more exercises to make the abdominal muscles strong. Thsi would help your situation.

Get a surgeon opinion n plan further treatment

A postpartum umbilical hernia (also called as “postpartum hernia”) is a postnatal condition that can commonly be coupled with diastasis recti and occurs when part of your intestine sticks out through a small tear opening in your weakened abdominal muscle wall . You could notice a protruding belly button, pain when core movements are performed or when pressure is placed on the area. It can also impede the total recovery of diastasis recti. It is not always necessary to resolve this with surgery as it is unique in the level of pain and severity of the condition for each individual.

Umbilical hernia ( belly button coming out ) during pregnancy is common due to stretching of umbilicus and muscle weakness ,But it reduces after the delivery (cesarean/vaginal ),but you might feel defect but there should not be any content (you should not feel anything within the belly button) , best is to get an ultrasound done to see any defect ,its size and content if any , If nothing comes out in Ultrasound then continue with excercise .

After pregnancy, especially after a C-section, it is common for the abdominal muscles and connective tissues to become stretched and weak. This can make the belly button look more prominent or ‘out’. If your doctor examined you and ruled out an umbilical hernia, and you have no pain, swelling, vomiting, or a lump that is increasing, it is usually not a dangerous condition. This may improve gradually over the next few months as the abdominal wall recovers. Gentle core strengthening exercises (as advised by your doctor/physiotherapist), good posture, and avoiding heavy lifting can help. Since you are only 3 months postpartum, recovery is still ongoing.

Based on your history, this is most likely due to stretching and weakness of the abdominal wall after pregnancy, with or without mild rectus diastasis. It is reassuring that you have no pain and your gynecologist did not find evidence of an umbilical hernia on examination. At present, in the absence of concerning symptoms, conservative management with observation and exercises is appropriate

As you have mentioned - cough impulse test and leg raising was done. Many times if the defect of hernia is very small it can be missed on clinical examination. Get USG abdomen done. Here, you can continue exercises as told by youtlr doctor to strengthen the abdominal muscles. If you still think that on coughing the umbilical site protrudes ---- go for re-examination along with USG abdomen.