If we plot the increase in Cl and the increase in Cd against increasing flap angle we will find that we get most of the Cl increase in the first few degrees of flap deployment and most of the increase in Cd in the last few degrees. This makes it tempting to suggest that increasing flap angle by a few degrees will increase lift:drag ratio, but in most cases it does not.
Under "normal circumstances" increasing flap angle will decrease lift:drag ratio. My reason for saying "under normal circumstances" is that there is at least one special case in which this is not true.
Imagine an aircraft for which the clean stall speed is 100 kts and the stall speed with full flap is 80 kts. What flap angle will give the best lift:drag ratio if we fly at 90 kts? It does not take too much thought to see that under these very special circumstances it is the full flap configuration.
Even under more normal circumstances the conventional wisdom can be incorrect. Most users of the 737 would argue that zero flap gives best lift:drag. But a recent article in Flight magazine reported a modification that reduces drag in the cruise by setting the flaps at a very small downward angle (less than 1 degree I believe). All this really means of course is that BOEING could have done better in designing the wing had they chosen to do so.