If the degree of S is 2, then take H a tangent plane, and the corresponding section is free.
When the degree of S is 3, then take H a tritangent plane, and the corresponding section is free.
First interesting case deg S=4.
The answer seems to be negative in general. Indeed a generic surface of degree at least 5 has only plane sections with total Milnor number at most 3, see
Proposition 4 in Geng Xu (J. Differential Geom. 39 (1994), no. 1, 139-172). Such curves are never free.
See for instance:
A. Simis, S.O. Toh\u aneanu, Homology of homogeneous divisors, Israel J. Math. \textbf{200} (2014), 449-487.
\bibitem{T} S. O. Toh\u aneanu, On freeness of divisors on $\PP^2$, Communications in Algebra, \textbf{41} (2013), 2916--2932.
The following is just a remark, not an answer (which you seem to have provided!). The question seems to be related to a question (of Ziegler, I think) for hyperplane arrangements, which was answered negatively by Yuzvinsky in "The first two obstructions to freeness of arrangements." That is, Yuzvinsky showed that a generic hyperplane section of any hyperplane arrangement is never a free arrangement (some mild assumption must be made on the original hyperplane arrangement - for instance it should not be a boolean arrangement). This is somewhat on the opposite end from a smooth surface, but perhaps a generic section of an arbitrary smooth surface satisfies the same conclusion.
Thank you for your comment and your question. Indeed, a generic plane section of a smooth surface of degree d has exactly one node, and hence is never free for d>2.
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