The answer is, indeed, "No" because there is an unbounded with probability $1$ stochastic process that satisfies the given inequality, namely, $X(t)=0.1\log|t-w|$ where $w$ is equidistributed on $[0,1]$. Truncating it at high level $L$, we get a continuous process such that $E|X(t)|$ is uniformly bounded but the supremum is identically $L$. Taking a suitable mixture of such truncations, we see that the tails may decay arbitrarily slowly.
There is essentially only one universal method to gets bounds for the supremum from the bounds for the increments, which is to consider $\delta_ k$-nets with diminishing $\delta_ k>0$ and bound the supremum by the convergent series of suprema taken over finite sets (differences between points from 2 successive nets). Clever choice of the nets may be crucial for the success but not in this case. Whatever you can get from the standard dyadic nets here is the best you can say.
I'm also tempted to ask whether you, indeed, need the estimate for the supremum (which cannot be made without extra assumptions) rather than for some $L^p$ norm.