I had a terrible bout of Achilles tendonosis after I ran a marathon in 2016. I attribute it to running in racing flats and starting my training back up too aggressively after the marathon. I have run one marathon since (in May) in which I PRd and also have had zero issues since. Before I give my opinion on the answer to your question, these are my keys to success:
1. Strength train if you don't already. As the other guy mentioned make sure to do eccentric heel drops (make sure not to overstretch). Don't forget to also incorporate glute and hamstring exercises (squats, leg press, hamstring curls, etc.). Weak glutes can cause you to overuse your calves and give you achilles issues. Once you are fully comfortable with eccentric heel drops start slowly incorporating weighted calf extensions (both standing and seated). The standing will get more of the gastronemius, etc. and the seated will work more on the soleus. I worked up to eight sets of six with heavier weights.
2. Foam roll like a MF. If your glutes and hamstrings are tight or knotted up, they will yank on your calf/ Achilles. Your recovery will take much longer if the muscles and tendons are not strong, loose and flexible.
3. If you work a desk job, get a standing desk. The more you sit, the more your hamstrings will tighten up and pull on other muscles. Its important to extend the muscles regularly.
4. Incorporate dynamic stretching (leg swings front to back, side to side). Do these several times a day. I do sets of twenty each.
5. Wear comfortable shoes as often as you can, when not running (wear comfortable shoes while running as well). My shoe of choice is the Adidas Ultra Boost when not running - this is the most comfortable shoe I've ever worn. I also wear recovery sandals in my house rather than walking barefoot -this is also key in my opinion.
6. Eat quality foods and try to avoid processed as much as possible. Any foods that are processed or have chemicals will only slow the healing in my opinion.
Now, to answer your question. My experience was that I still have a little soreness from time to time in the Achilles after tough runs (but only when I squeeze it). It never hurts during the run, and it seems to always go away by the next day. I'd say if you are still tight when you wake up in the morning, you have more work to do before you are ready to start running again. The last thing you want to do is to jump back in too soon. I recommend following the steps I layed out above (maybe get in the gym more, swim, etc.), until you are no longer sore in the mornings. Then slowly start getting back into running.
I hope that helps.