I do not understand why you have 30Hz resolution so I will focus only to the principle of the question "does interpolation increase resolution?".
Short answer is no, no new data, no new information.
A longer answer needs the spectrum visualization below, with time domain, continuous frequency domain and discrete frequency domain from left to right.
The interpolation technique here is to preserve the information of spectrum. DFT works on discrete frequency domain which is the part from $-f_s$ to $f_s$ of the continuous frequency version.
First, look at the continuous frequency domain, if you upsample your signal correctly, it is equivalent to changing the sampling frequency. You wish to double the number of data, but it is just removing one-half the spectrum replicas of sampling process.
Now, look at the discrete frequency one. This version is normalized from $-f_s$ to $f_s$ of the continuous frequency counterpart. $f_s$ is doubled, the spectrum is then shrinked by a factor 1/2. If we call $0 < \alpha < 1$ the proportion of non-zero frequencies before upsampling, this proportion is $\alpha/2$ after upsampling. Before upsampling, DFT gives you $N$ bins for $\alpha$ then $N\alpha$ bins for the spectrum; after upsampling it is $2N$ for $\alpha/2$ then always $2N \times \alpha/2 = N\alpha$ bins for the same spectrum. No, your resolution does not change at all.
To have smoother resolution, the only way is to add more data. In your example, instead of dividing to 30 frames, divide your audio file to 15 frames.