Homomorphic/Paillier crypto system for use case?: overflow for multiple counter exponent possible? Different cipher factor needed all the time?
It won't work anyways because even in case it would be secure with one $r$ and the exponent can get overflow for each candidate the ciphertext with same exponent will be different after an overflow happened. 


  Do you really mean deterministic (that is, one particular m always maps to the same c)   


Given two ciphertext with same exponent ($\mod n$) before and after overflow. The 2nd would be result after adding $n$ times one vote to first.
$c_0 = g^0 \cdot r^n \mod n^2$
$c_{i+1} = c_i \cdot c_{add+1} = c_i \cdot g^1 \cdot r^n = g^i \cdot r^{in+n}$
$c_a=g^a \cdot r^{an+n} \not=g^{a+n} \cdot r^{an+n^2+n}=c_{a+n}$ $\mod n^2$
(because $r^{\phi(n^2)}=1$ and $n^2 \not = \phi(n^2)$  )
Decryption resolve in same $a$ but exponent of $r$ is different and with this those two ciphertexts.