UDP-3-O-(R-3-hydroxymyristoyl)-N-acetylglucosamine deacetylase (LpxC) is a zinc amidase that catalyzes the second step of the biosynthesis of lipid A, which is an outer membrane essential structural component of Gram-negative bacteria. Inhibitors of this enzyme can be attributed to two main categories, non-hydroxamate and hydroxamate inhibitors, with the latter being the most effective given the chelation of Zn2+ in the active site. Compounds containing diacetylene or acetylene tails and the sulfonic head, as well as oxazoline derivatives of hydroxamic acids, are among the LpxC inhibitors with the most profound antibacterial activity. The present article describes the synthesis of novel functional derivatives of hydroxamic acids—bioisosteric to oxazoline inhibitors—containing 1,2,4- and 1,3,4-oxadiazole cores and studies of their cytotoxicity, antibacterial activity, and antibiotic potentiation. Some of the hydroxamic acids we obtained (9c, 9d, 23a, 23c, 30b, 36) showed significant potentiation in nalidixic acid, rifampicin, and kanamycin against the growth of laboratory-strain Escherichia coli MG1655. Two lead compounds (9c, 9d) significantly reduced Pseudomonas aeruginosa ATCC 27853 growth in the presence of nalidixic acid and rifampicin.