Is it possible to generate Hamming code with minimum distance $d$ which can correct more than 1 bit error?
You cannot detect 2-bit errors (reliably) with a Hamming code distance of 3 if you are correcting 1-bit errors.

At any rate, you can design Hamming codes correcting two errors.  Let's assume 32bit of data and n Hamming code bits.  The states we then need to be able to encode is
0 bit error = 1 state
1 bit error = 32+n states
2 bit error = (32+n)(31+n)/2 states
For n=10, this amounts to 1+32+10+21*41 = 904 states that can be encoded with 10 bits of redundancy.  Depending on how complex you want your correction machinery to get, you might want to use more bits in order to make multiple error correction less complex.