Looking for cycles of periods longer than the input signal length
If you plot frequency instead of period, you might be able to see that some of the increasing artifacts are due to the shrinking rectangular window on your signal.  

An FFT decomposes the entire length of its input into sinusoids.  You have created a flat spot in your data input, and it takes a linear combination of a lot of sinusoids to represent those extended flat spots.  So the FFT of your latter shorter signal segments becomes more and more about the flat spot than your signal.

Added: There are other methods for estimating the parameters of a known small number of sinusoids within a known distribution of noise, such as MUSIC and ESPIRIT, that may be more suitable for estimation of sine waves not equal to or possibly shorter than the lowest DFT basis vectors.