We report magnetization, heat capacity, thermal expansion, and magnetostriction measurements down to millikelvin temperatures on the triangular antiferromagnet YbMgGaO$_4$. Our data exclude the formation of the distinct $frac13$ plateau phase observed in other triangular antiferromagnets, but reveal plateaulike features in second derivatives of the free energy, magnetic susceptibility and specific heat, at $mu_0H$ = 1.0 - 2.5 T for $Hparallel{}c$ and 2 - 5 T for $Hperp{}c$. Using Monte-Carlo simulations of a realistic spin Hamiltonian, we ascribe these features to nonmonotonic changes in the magnetization and the $frac12$ plateau that is smeared out by the random distribution of exchange couplings in YbMgGaO$_4$.
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