FFF, if you're maintaining level flight on the back of the power curve and then reduce power your speed will reduce, not increase - leading to a further & continuing speed loss. Thrust is being used to counter Drag. Remove any amount of Thrust & the now excess Drag will cause a further speed loss. You would have to lose altitude to maintain the speed in this case.
The point about flight on the back side of the curve is that the a/c is no longer speed stable. A speed loss results in more drag and a further speed loss, a speed gain results in less drag & a further speed gain. That's why that part of the curve is called the 'region of reversed command'. On the normal - or front - side of the curve additional speed results in more drag. If thrust hasn't been changed then the additional drag acts to reduce the a/c's speed back to the point where Thrust = Drag.
No matter which side of the curve you're on, less Thrust will result in a speed loss and more Thrust is needed to gain speed. The only options are to lose altitude ie convert height into speed, or reduce drag eg change from a higher drag configuration to a lower drag config.
WRT to your a/c's speed gain on a power reduction, I can think of at least two reasons:
1. A change in airflow around the static &/or pitot sources causing an error.
2. If the a/c is allowed to pitch ND following the power reduction then it will normally pitch further than the steady state pitch attititude needed to retain equilibrium. In doing so its speed will increase beyond the steady state 'on trim' speed. As a result the a/c will pitch NU, then ND etc etc in a diminishing oscillation until it eventually regains equilbrium.