Can I take piecewise FFT of a larger input sample?
You need the Cooley-Tukey algorithm, which can be used to express the DFT of size $N=N_1N_2$ by DFTs of sizes $N_1$ and $N_2$. It works a bit like what you sketched in your question, but you forgot the twiddle factors. This answer explains the details of the algorithm and it also includes a simple Matlab implementation.

EDIT (see comments): If you really want to compute the FFTs of blocks of consecutive data you need to split the computation of even and odd frequency indices (assuming the DFT length $N$ is even):

$$X[2k]=\sum_{n=0}^{N/2-1}\left(x[n]+x[n+N/2]\right)W_{N/2}^{nk}\\
X[2k+1]=\sum_{n=0}^{N/2-1}\left(x[n]-x[n+N/2]\right)W_N^nW_{N/2}^{nk}\tag{1}$$

where $W_N=e^{-j2\pi/N}$. Note that you can of course compute the DFTs of the length $N/2$ sub-blocks separately instead of adding (or subtracting) the blocks before the DFT (as in (1)), but this is computationally less efficient.