Structure of the periodic table The periodic table I understand is arranged by order of increasing atomic number after Henry Moseley's X-ray experiment. 

But I also learned that the periodic table also has its shape due to the types of subshells present in certain sections called blocks of the table.

However, I still do not understand who shaped the modern periodic table so it mimics the filling of lattermost subshells of atoms of each element (in order of increasing number of electrons- from left to right as well as top to bottom for the elements.)   

My Question is : was the periodic table Mendeleev and Moseley devised not in a shape that can be used to predict the filling order of each elements, and that the periodic table was later adapted to fit this concept of orbitals in?
Mendeleev was the first to realize that properties recur among the elements. To highlight this, he devised the early periodic table based on atomic masses. Moseley came along and stated that the periodic properties of elements was a function of their atomic numbers rather than their atomic masses. 



It was Niels Bohr who came up with the idea of shells and subshells. He developed his own take on the periodic table, which looked somewhat like this:



Finally, coming to your question, A lot of derivatives of the periodic table were made by various scientists during the 1920's - 1940's. The version we use today was made by Glenn T. Seaborg and his colleagues.


  Seaborg and collaborators had synthetically produced several new elements with atomic numbers beyond uranium, the last naturally occurring element in the table. Seaborg saw that these elements, the transuranics (plus the three elements preceding uranium) demanded a new row in the table, something Mendeleev had not foreseen. Seaborg’s table added the row for those elements beneath a similar row for the rare earth elements, whose proper place had never been quite clear, either.


Further reading:


How the periodic table went from a sketch to an enduring masterpiece
Further development of the Periodic Table - Wikipedia