How much more secure is $c = mg_1^r + g_2(g_1(g_1^r-1)/(g_1-1)) \mod p$ compared to just $c = mg_1^r \mod p$ (dis. log), all known but $r$?
If the attacker knows everything (including $m$) except for $r$, he can rewrite it to:

$$g_1^r = (c + g_1 g_2 (g_1 - 1)^{-1}) ( m + g_1 g_2( g_1 - 1)^{-1})^{-1} \pmod{p}$$

He knows everything on the RHS, and so it's a simple discrete log problem.

BTW: for the $\mathbb{Z}_p^*$ group, there are much faster attacks than Baby-step-Giant-step