With a constant current source, you cannot make it constant current and constant voltage, if the load changes and the current does not, the voltage must change. This is due to Ohms law:

V = I R

If I is to remain constant, and you change R then V must change in the same direction. I.e. if you double R then you will have to double V to get the same I.

Yes, I'm using a zenerdiode for the constant reference drop. The problem is exactly like you are describing Bob. I get a voltage change when change the load, but I'm thinking for a way to prevent that or adding another circuit to get the voltage back.

I tried to get the same voltage back by amplifying it, but it didn't worked out.

You cannot have a constant current AND a constant voltage with a changing load. This is basic electronics (Ohms law) as I already showed you. If the load changes, the voltage must change to keep the current constant. A constant current source could be described as a circuit that varies the voltage in order to keep the current constant.

When you say you get a voltage change when you change load, that is to be expected -- it is the function of a constant current source.

Does the current vary significantly?

With a good constant current source the voltage should range from close to zero to close to the supply rail as the load varies. Only once the constant current source is no longer able to raise the voltage higher should the current start to fall significantly.

The current doesn't change and the voltage only drops around 1 volt, like expected.

So I thought I could try the non-inverting amplifier to get more voltage at the output, but perhaps you have some more ideas I could do some research about.

I know I can't prevent the voltage drop of 1 volt. Is there a way of perhaps adding an extra circuit at the output of the constant current source to get the 1 volt back?

OddOne, what you're asking for can probably be best described using an analogy:

I have a cruise control on my car but I find that with it turned on, the engine throttles back when I am going downhill. Is there any way I can retain the constant speed, yet go faster down the hills?