I would recommend that you look at two different aspects of your experimental conditions to begin with:
The first thing is to reduce the secondary antibody concentration sequentially in order to see if that reduces the background.
Secondly, if you are using HRP detenction method then you can reduce the time of exposure of your blot to the HRP substrate because sometimes overexposure of the blot to the substrate leads to a very noisy background during detection.
Another option is to block with higher % of BSA or milk or simply load less sample. We also often use different dilutions of primary antibody, or even dilute them in milk instead of TBST. All these tricks are basically just hiding noise by decreasing overall sensitivity. Question is, is your background noise of the same intensity as the band of interest? If antibody is not commercial and you have enough of it you could also try to affinity purify it, by running over a column with immobilized epitope, but that is quite tricky and I am not sure, how desperately you need to clean up the signal. I also tried cleaning up the antibody milk by incubating it with a membrane loaded with samples from deletion strain that do not contain the protein of interest.