In an ideal world, if the characteristic impedance of your transmission line matches the system impedance (usually 50Ω), then adding or removing lengths of transmission line just moves the complex impedance around a constant-SWR circle. So you may get a different impedance (it may look capacitive instead of inductive, etc), but the SWR will always be the same, regardless of how much transmission line is between your meter and the antenna.
However, the real world is not ideal. Firstly, if the transmission line is lossy, that will make the SWR better. It does so because the transmission line absorbs some of the transmitted energy, and then absorbs again the reflected wave. No reflected wave means 1:1 SWR. Absorbing all the energy is one way to reduce the reflected wave.
Secondly, there are a great many amateur antenna designs which, intentionally or not, use the feedline as part of the antenna. These antenna currents on the feedline are called common-mode currents. When they exist, the feedline is very much part of the antenna, and changing the feedline length can change the impedance of the antenna, just like changing the length of any of the other antenna elements would. For this reason and many others, good antenna designs do not have significant common-mode currents.