Does amp output V decrease by adding gain stage?

My tube amplifier is low sensitivity, has high output voltage at 4V. I plan to add one or two gain stages to improve the amp sensitivity. The question is by adding gain stages will that results in lowering output voltage?
Increasing gain in the input voltage amplifier stages will increase input sensitivity but should have no effect on the capabilities of the output stage.

But a 4V output for an amplifier is very, very low. Perhaps you are talking about a preamp?
I made a terrible mistake. The INPUT voltage is 4 voltage. Very insensitive amps I got to say
You can add as much gain as you want before or at the input of the amp but it will not change the amps output in any way. IOW whatever it is rated at in terms of full output wattage will be the same. What you will change is the sensitivity of the amp. IOW you won't need 4 volts to get the amp to full output.
Is that the input sensitivity? Balanced or RCA? You might need one hulk of a preamp.
That is RCA input. So the amp can mate with a giant preamp to take at least 4 Volt out to drive the amp. I am thinking to put gain stage(s) in the input of the amp. So the question is if I add gain stage(s) to the amp, does it mean the input voltage of the amp will reduce below 4 Volts?
Yes, adding a gain stage at the input would reduce the voltage needed. How much less voltage depends on the amount of gain that you add.
To get to a 1 volt level you will need to add 12db, which is easy with most tubes that might be in a gain stage.
Is your tube amp a "pull" from a console that was wired for Stand Alone?
I have an RCA 6bq5 amplifier that sounds much like yours from your description. The matching preamp from the console was very high output.
Although I haven't gotten around to it, I see no reason why I can't change the input stage and feedback loop to make it louder (more gain).
If your amp is in fact a "pull", you're going to need a schematic with voltages. For example, you can't simply reduce the amount of negative feedback without the risk of causing oscillation or motor boating.
You might want to join one of the DIY forums for more information. That's what those guys do over there, make stuff work.
I have a few different "pulls" and it's not simply adding an on/off switch and mounting jacks. My Fisher 481-A amplifier chassis also powered the preamp and multiplexer before it was removed. To run it as a stand alone, I had to add resistors to make up for the missing preamp/multiplexer. Otherwise, the voltages were too high in the amp.
This can be a great learning experience for you if you have the inclination and desire. I loved my Fisher 481 to death before I got a Fisher SA100, which is what I'm listening to right now. The RCA went to the back of the line because I have six other amps I want to fix first. But I see no reason why I can't make it more sensitive.

What kind of amp are you talking about anyway??