There are many "natural" or "clean label" ingredients that work as mold inhibitors. Most of them work by adding some acid and thereby creating a more inhospitable environment for mold. Some of the popular mold inhibitors (at least is some U.S. bakeries) are: cultured whey, vinegar, and raisin juice.
I think vinegar and raisin juice are good options for home use.
Works by adding Acetic acid. The white vinegar you buy in the grocery store is typically 5% acetic acid (at least in the U.S.).
To inhibit mold, the recommended amount is 0.5-2.0% of the flour weight of 5% acetic acid vinegar. Add with your other liquids.
Raisin Juice Concentrate
Raisin juice concentrate has been shown to be effective at retarding mold and bacteria growth in bread. It works by adding Propionic acid and Tartaric acid. Further, it seems that a home baker should be able to purchase this in reasonable quantities and at a reasonable price.
The recommended amount is 5-10% of flour weight, by weight. Add with your other liquids.
My personal experience is positive: I used to purchase a whole-wheat bread with raisin juice concentrate and it lasted forever.
This study from the Journal of Food Science, Application of raisin extracts as preservatives in liquid bread and bread systems, found very dramatic increase in shelf life.
The mean mold-free shelf life of the bread containing 7.5% water extract [of raisins] was 18.1 +/- 3.3 d at room temperature while the negative control was mold free for 9.4 +/- 2.4 d. The antifungal efficacy of the extracts in bread was equivalent to 0.24% calcium propionate in 21 d of storage. Doubling the concentration of the extract did not improve the mold-retarding property in bread. The bread containing raisin paste, the percentage of which in dough was equivalent to 15% raisin extract, exhibited a stronger antifungal activity than did the extracts in bread.
Both of these can inhibit yeast activity which you may need to compensate for by increasing yeast or proofing time.
Sources: some personal experience, this really interesting article from Oklahoma State University, Clean Label Mold Inhibitors for Baking, and the study linked above.