Brute forcing an aperiodic s-shaped signal to be periodic by mirroring it before performing FFT
Cascading the initial signal with it’s reverse does two notable things which facilitates the OP’s further processing:
First and most significantly it minimizes the end discontinuities which are visible by creating the equivalent periodic waveform. The Fourier Transform result for the finite duration waveform is equivalent to the transform for the same waveform periodically extended for all time (the non-zero Fourier results specifically). This visualization is useful to more intuitively understand the effect of the discontinuities in an equivalent continuous-time sampled context: very high frequency components are needed to transition in a very short period of time; these high frequency components alias back to lower frequencies for a sampled system and we can see the result as “spectral leakage”).
Second, as a note of interest, it creates a linear phase response in frequency. This is similar but not identical to the cascade of any FIR filter with its reverse filter: the resulting coefficients after the convolution of the two will be symmetric and the frequency response will be linear phase with a magnitude response as the cascade of the two filters (thus having a “squared magnitude” response).  Consider if t=0 were at the center of the resulting mirrored time domain signal: the resulting frequency domain signal as the Fourier Transform would be real (symmetric about the vertical axis at t=0 in the time domain, or f=0 in the frequency domain will always be real in the other domain). By then shifting it to a later time as a causal signal, instead of being real in frequency, the resulting response with the exact same magnitude has an additional linear phase added but is otherwise not changed (a linear phase in frequency is a fixed delay in time). An example of this that may be more intuitive is a similar example in the frequency domain and how the time domain would be real because of the symmetry: Consider $2\cos(\omega t)$ which is a real time domain signal and has two symmetric impulses in the frequency domain at $\pm \omega$, (and we see this with Euler’s formula with $2\cos(\omega t)= e^{j \omega t} + e^{-j \omega t}$). With just one of those frequency domain impulses at $+\omega$, the time domain waveform would be complex and given as $e^{j \omega t}$.
This second point is an outcome of the theorem that the Fourier Transform of a real valued sequence is conjugate symmetric (so equivalently the Fourier Transform of a conjugate symmetric sequence is real), and is just describing how the result becomes linear phase (real plus an additional time delay so then complex with linear phase). Most significantly the OP has reduced the aliasing effects from the otherwise very large discontinuity in time while the original time domain sequence is completely recoverable from the frequency domain, so less prone to distortion effects with any subsequent processing in frequency prior to transforming back to the time domain, and noted that this was done without modification to any of the original time domain samples (as would be done with windowing). This was feasible since the derivative of the original waveform was zero or close to zero at the edges, thus minimizing the discontinuity in the equivalent periodic waveform after adding the mirrored component.