How about this:
$$\partial_{a'}T_{b'_1b'_2}=\left(\partial_{\sigma'}T_{\mu'_1\mu'_2}\right)dx^{\sigma'}\otimes dx^{\mu_1'}\otimes dx^{\mu_2'}=\left(\partial_{\sigma'}T_{\mu'_1\mu'_2}\right)\frac{\partial x^{\sigma'}}{\partial x^\sigma}\frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\frac{\partial x^{\mu_2'}}{\partial x^{\mu_2}}dx^{\sigma}\otimes dx^{\mu_1}\otimes dx^{\mu_2}$$
Now to deal only with components:
$$
\left(\partial_{\sigma'}T_{\mu'_1\mu'_2}\right)\frac{\partial x^{\sigma'}}{\partial x^\sigma}\frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\frac{\partial x^{\mu_2'}}{\partial x^{\mu_2}} = \frac{\partial x^{\sigma'}}{\partial x^\sigma}\partial_{\sigma'}\left(T_{\mu'_1\mu'_2}\frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\frac{\partial x^{\mu_2'}}{\partial x^{\mu_2}}\right)-\frac{\partial x^{\sigma'}}{\partial x^\sigma}T_{\mu'_1\mu'_2}\partial_{\sigma'}\left(\frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\frac{\partial x^{\mu_2'}}{\partial x^{\mu_2}}\right)=\partial_{\sigma}T_{\mu_1\mu_2}-T_{\nu_1\nu_2}K^{\nu_1\nu_2}_{\sigma\mu_1\mu_2},
$$
where:
$$
K^{\nu_1\nu_2}_{\sigma\mu_1\mu_2}=\frac{\partial x^{\nu_1}}{\partial x^{\mu_1'}}\frac{\partial x^{\nu_2}}{\partial x^{\mu_2'}}\partial_{\sigma}\left(\frac{\partial x^{\mu_1'}}{\partial x^\mu_1}\frac{\partial x^{\mu_2'}}{\partial x^\mu_2}\right)=\frac{\partial x^{\nu_1}}{\partial x^{\mu_1'}}\frac{\partial x^{\nu_2}}{\partial x^{\mu_2'}}\frac{\partial x^{\mu_2'}}{\partial x^\mu_2}\frac{\partial^2 x^{\mu_1'}}{\partial x^{\mu_1}\partial_\sigma}+\frac{\partial x^{\nu_1}}{\partial x^{\mu_1'}}\frac{\partial x^{\nu_2}}{\partial x^{\mu_2'}}\frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\frac{\partial^2 x^{\mu_2'}}{\partial x^{\mu_2}\partial_\sigma}=
\frac{\partial x^{\nu_1}}{\partial x^{\mu_1'}}\frac{\partial x^{\nu_2}}{\partial x^{\mu_2}}\frac{\partial^2 x^{\mu_1'}}{\partial x^{\mu_1}\partial_\sigma}+\frac{\partial x^{\nu_1}}{\partial x^{\mu_1}}\frac{\partial x^{\nu_2}}{\partial x^{\mu_2'}}\frac{\partial^2 x^{\mu_2'}}{\partial x^{\mu_2}\partial_\sigma}=\frac{\partial x^{\nu_1}}{\partial x^{\mu_1'}}\frac{\partial^2 x^{\mu_1'}}{\partial x^{\mu_1}\partial_\sigma}\delta^{\nu_2}_{\mu_2}+\frac{\partial x^{\nu_2}}{\partial x^{\mu_2'}}\frac{\partial^2 x^{\mu_2'}}{\partial x^{\mu_2}\partial_\sigma}\delta^{\nu_1}_{\mu_1}\equiv M^{\nu_1}_{\sigma\mu_1}\delta^{\nu_2}_{\mu_2}+M^{\nu_2}_{\sigma\mu_2}\delta^{\nu_1}_{\mu_1}
$$
so it holds:
$$
\partial_{a'}T_{b'_1b'_2}=\partial_{a}T_{b_1b_2}-\left(T_{\nu_1\mu_2}M^{\nu_1}_{\sigma\mu_1}+T_{\mu_1\nu_2}M^{\nu_2}_{\sigma\mu_2}\right)dx^\sigma\otimes dx^{\mu_1}\otimes dx^{\mu_2} = \partial_{a}T_{b_1b_2} - T_{b_1c} M^c_{ab_2}-T_{cb_2} M^c_{ab_1},
$$
where we defined tensor field:
$$
M^a_{b_1b_2}\equiv M^{\nu}_{\sigma\mu}\partial_{\nu} \otimes dx^\sigma\otimes dx^{\mu}\equiv \frac{\partial x^{\nu}}{\partial x^{\mu'}}\frac{\partial^2 x^{\mu'}}{\partial x^{\mu}\partial x^\sigma} \partial_{\nu} \otimes dx^\sigma\otimes dx^{\mu}.
$$
For higher order covariant tensor fields we would get:
$$
\partial_{a'}T_{b'_1b'_2..b'_n} = \partial_{a}T_{b_1b_2} -T_{cb_2..b_n} M^c_{ab_1}- T_{b_1cb_3..b_n} M^c_{ab_2}-...-T_{b_1..b_{n-1}c} M^c_{ab_n}
$$
In full analogy you can check how the operator acts on contravariant indexes.
Note however, that tensor field $M$ is indeed a tensor field - that is it transforms under change of coordinates in the correct way. However the meaning of the tensor is relating the operators $\partial_a$ and $\partial_{a'}$. The components of the tensor in different coordinate system would not relate derivative operators associated with these new coordinate system, rather the components would still relate the old derivative operators, albeit expressed in different coordinate system.
