As long as an object has positive acceleration, its speed will be increasing.
If an object is moving at, say 100 m/s, with acceleration, 10 m/s^2, as the acceleration decreases to 5 m/s^2, that is still a positive acceleration so its speed is still increasing.
Think of being in a fast car getting onto the interstate from an on ramp. If you floor the accelerator, you will accelerate pretty quickly. Now, if you let off the accelerator a little bit, your acceleration slows down, but you are still increasing your velocity. Until the acceleration reaches zero or goes negative, then you will increase in speed.