An excellent question!
Producing steam is converting a liquid to a gas and would involve a a large volume expansion. When you talk about evaporation and boiling, you are talking about two different processes. You can read more about the different between the two here

Evaporation occurs only at the surface and happens at any temperature. For any temperature below 100 degrees Celsius and at 1 atm, higher temperatures will lead to more evaporation and some expansion of volume. However, the volume expansion is not as large as converting water to steam. Evaporation is also counterbalanced by condensation and is dominant at temperatures below the boiling point. As it occurs mostly at lower temperatures, it would not lead to as great a volume expansion.

We can understand this first part of volume expansion upon boiling by thinking of steam (a vapor at high temperature) as an ideal gas (not a totally correct picture, but it will illustrate how to think through the problem). The ideal gas law can be derived using thermodynamics and has the form PV = nRT, where P is the pressure, V is the volume, n is the number moles of the species you have, R is gas constant, and T is temperature. What is ideal gas law tells you pressure and volume are inversely proportional, but pressure and volume are proportional to temperature. For example, the same amount of gas (in moles) n in a smaller container will result in a larger pressure. Another example is a higher temperature will increase the kinetic energy of the gas particles and cause them to collide with the sides of the container more frequently and at a higher velocity, which corresponds directly to a pressure. Similarly, keeping the pressure constant would require a larger volume container for increasing temperature.

More generally, the state of water depends on both temperature and pressure. The common temperatures we know ice and vapor to form are only true for atmospheric pressure, which is the pressure most everyday things happen.

Phase diagram of water
People have figured out using thermodynamics the dependence of phase (what you call the state of water) on temperature and pressure, and put them on graphs called phase diagrams. A phase diagram for water can be found here

By picking a particular temperature and pressure, you can figure out what phase water is.

If you are interested in what the different features of the phase diagram means, you can read more about it here

Basically, the lines that separate the regions of ice, water, and vapor are lines of two-phase coexistence. That means at temperatures and pressures along the boundary between water and ice, both water and ice can be found at the same time with each other. The triple point is where all the phase boundaries intersect and marks a special temperature and pressure where all three phases exist in equilibrium.

Answering the last question, the phase you get really depends on which temperature and pressure you are in. For example, we generally associate the boiling temperature of water to be at 100 degrees C, but this is specific to a pressure of 1 atm (i.e., the pressure around sea level). If you were at higher pressures, you would need higher temperatures to boil the water. This is the idea behind pressure cookers, for instance.

Hope this helps!
Best,

Water when it evaporates has no set new volume. It becomes a gas, like air. In air pressure, water reaches the same density in molecules/volume as air does (air is mostly nitrogen). The less dense the water vapor, the more it expands.

Water can remain liquid at higher temperature than it can under atmospheric pressure, and boils at lower temperature at lower pressure. For example, on Mars, the pressure is so low that water boils even at its freezing point. There is a point above which water never remains liquid, however, and enters a phase called a supercritical fluid, which is like a very dense (and non-ideal) gas.