reading different files from the subroutine depending on the input

The length of the filename array is given as (m), which is the dummy argument for which of the files you want to read.

So if, for example, you call My_Reader(3), it will only initialize a 3-element array for filename and then anything can happen when you write something to the 6th element.

You could simply fix the size of the filename array in the subroutine declaration block:

character(len=256) :: filename(6)

but I would do something completely different, I'd use a select case to assign the filename in the subroutine:

subroutine my_reader(m)
    integer, intent(in) :: m
    character(len=256) :: filename

    select case (m)
        case(3)
            filename = 'C:\Users\spaudel\Documents\S3.txt'
        case(6)
            filename = 'C:\Users\spaudel\Documents\S6.txt'
        case default
            print *, 'incorrect selection of file number: `, m
            STOP
    end select

    open(unit=m*10, file=filename, ...)
end subroutine my_reader