When to take baby to doctor?

My little guy is 9.5 months and he’s been sick for probably close to a week. For the first few days he was hot and grumpy and had a hard time over night. Then a day went by and it seemed like he got better and now he’s developed congestion and a cough that sounds like he has stuff in his throat.

We’re all sick too and he’s having the same symptoms as us. He’s not acting sick other than the cough and congestion. He feels a little hot but not as bad as the first few days. We haven’t taken his temp. Also he did have the flu vaccine if that matters but I don’t think he has the flu.

Should I schedule him a doctor’s appointment? I haven’t because he obviously just has the same cold as the rest of us (although none of us have had a fever) but I know sickness is different with babies than adults.