Heat transfer dependence on temperature gradient?
Yes, it is very much dependent on temperature gradient. The stefan-boltzman law is only for radiative dissipation of heat and not for general heat transfer by conduction(contact).
The equation for steady state heat flow is
$$\vec h=-k \vec{\nabla T}$$ where $\vec \nabla$ is the gradient operator which is based on the temperature difference between the bodies and $k$ is the thermal conductivity. $\vec h$ is the heat per unit area.  

If the system is not in steady state, then the heat diffusion equation becomes relavent:- 
$$\frac{dT}{dt}=\frac{k}{c}\nabla^2 T$$ where $c$ is the specific heat of the body.
Lastly, a to a crude approximation, the rate of cooling in atmosphere at low difference of temperature of atmosphere and the body is given by Newton's law of cooling (partly by radiation):- 
$$\frac{dT}{dt}=-K(T_{body}-T_{surrounding})$$ where $K$ is a constant.   

In all these equation, it is the temperature gradient between the bodies in contact, which drives the rate of heat transfer (or temperature change).