The answers by Andy Walls and Matt L. are good answers and address many aspects of the question. They don't address the idea that I think was intended by the OP (and I might be mistaken). I believe the OP wants to zero pad the filter in a way that maintains the odd symmetry of the coefficients (presumably because the OP has a library function that speeds up the computation of odd symmetric filters). Algorthms that exploit odd symmetry result in subtracting two mirrored sample delays and multiplying them by the same coefficient, then summing up all of these products. This requires (about) half the number of multiplications as the naive approach.
As noted by the others, the magnitude response of all the filters will be the same no matter where you place the extra zeros. The phase response of these filters will all be different (equivalent up to a linear phase shift). However, you cannot maintain the odd symmetry of your odd length filter by zero padding it to an even length. This is because an odd length filter with odd symmetry has a single coefficient that is at the center (the odd symmetry is taken about this point). An even length filter with odd symmetry has no single coefficient at its center (the odd symmetry is taken about a point halfway between the two middle samples).
That's the bad news. The good news is that the center coefficient of a filter with odd symmetry is usually zero. So, if you have access to the "fast" algorithm, just delay the samples at the mirror-point by one additional delay prior to subtracting them and performing the chain of multiply-accumulate operations. This should be fairly easy to accomplish if you have access to the code. If not, then it probably isn't an option.
In general though, I'd do as Matt L. has suggested and simply redesign a filter with 1200 taps so that you can take full advantage of your processing, unless you have a good reason not to do so.