DIMAP/NODE contains 3 preposition senses for across; ODE contains only 2. DIMAP/NODE sense 2, “expressing movement over a place or region,” seems to have been absorbed into the main sense, “from one side to the other of . . .” Oxford treats the prepositional and adverbial use of across under the same senses because they are more or less inseparable. Following normal parsing conventions, across becomes a preposition when a complement is present, and is classed an adverb when one is not, even though one may be implied.
Other dictionaries give a more fine-grained treatment to across, which I think is in fact desirable. Hence, added senses. Sense 2 a bit lumpy . . .
Notes from the spreadsheet:
New sense added: (Spreadsheet record 3) ODE makes no provision for a common use of into that is treated in other dictionaries. See, e.g., sense 3 in MW11 ("used as a function word to indicate a period of time or an extent of space part of which is passed or occupied *far into the night*") and sense 8 in Random House Unabridged " (used to indicate a continuing extent in time or space): lasted into the night." SRtype: PeriodEntered. Definition: “used to indicate a period of time begun while some activity continues.” I subsumed it here because the period of time in some sense "engulfs" the thing that appears in it. Even though there are very few FrameNet instances, those that exist cannot be classified elsewhere.
Other instances of across that may deserve idiomatic treatment
ODE has the following: (1) across the board which is probably best singled out as a phrase. (2) across from, which you don’t identify as a preposition in DIMAP but should probably be treated as one. I’m not sure why ODE separates it, as it is pretty well subsumed under sense 3. Note that it is synonymous with opposite (sense 1 only), which you do identify as a preposition.
Across is the particle in a two groups of transitive phrasal verbs that share roughly the same meaning: (1) the inseparable come across, run across, and stumble across (“I came across these old photos today.”) and (2) the separable get across and put across (“I couldn’t get across the idea that it was important.”)
ODE’s two senses for across are necessarily lumpy; most other dictionaries have four main senses, and some subsenses. I debated whether to expand the ODE inventory further but have opted not to, on the assumption that standing divisions are, at present, fine-grained enough for our purposes. I would note however, concerning sense 1: we include in it a “dispersal” sense of across that is to some degree a figurative extension. In other words, if “from one side to another,” then effectively “in all parts of.” This usage is particularly common with verbs of dispersal (scatter, spread, splash, strew).