Can we define a new unit instead of specific heat that is constant with temperature rise?
There isn't a simple redefinition that would work in all cases, and this is probably why nobody does this.  Of course, you COULD define such a unit for a given system by defining it as a function of temperature, but it would be applicable only to the specific system.

As you noted, the physics behind the temperature dependence of a material's heat capacity lies in the quantum states which become available for excitation at different energies.  Consider an insulator which stores heat primarily in the lattice vibrations.  As you increase the temperature, more vibrational modes are accessible.  This is often described using the Debye model which gives the heat capacity as such:

$C_V = k(\frac{T}{\theta})^3$

where $C_V$ is the heat capacity, $k$ is a constant, $T$ is the temperature in Kelvin, and $\theta$ is a constant pertinent to a specific solid.[1]  

However, a metal can also store a significant amount of heat in the motions of electrons since conduction electrons are not bound to specific sites.  Heat capacity for the electron 'gas' is often described like this:

$C_V' = k' \frac{T}{\theta'}$

where $C_V' \neq C_V$, $k' \neq k$ and $\theta' \neq \theta$.[2]  Depending on $\theta$ and $\theta'$ for a given material, at low temperatures (often cryogenic) the electron contribution can be the dominant mode of storing heat in the solid.  Thus, if you chose a unit which was sensible for an insulator, it wouldn't apply to a conductor (and vice versa).  

Even within a specific material, the heat capacity on either side of a phase transition is often different since the phase transition has changed the quantum states available in the material.  Therefore, your new unit would become discontinuous at phase transitions and would need a different mathematical expression for each segment of the heat capacity function, not to mention a different parameterization for each material.  Hence, no real gain.

[1] See Kittel, Kroemer, Thermal Physics Second Edition; W. H. Freeman and Company, New York, 1980, p. 106, eq. 47b

[2] See Kittel, Kroemer, Thermal Physics Second Edition; W. H. Freeman and Company, New York, 1980, p. 193, eq. 37.  Note that folks don't actually use the symbol $\theta'$ in this case, but choosing so makes the explanation above clearer.