A verb appearing in The Etymologies of the 1930s as ath-rado “to cross, traverse”, clearly a combination of N. ath- “cross” and N. rada- “make a way” (Ety/RAT).
Neo-Quenya: I recommend against using the prefix N. ath- “cross” since its etymology seems to have been abandoned, but I would retain athrada- “cross, traverse” for purposes of Neo-Sindarin as an elaboration of athra- “cross (to and fro)”. I would use it for crossing in a single direction, as in i adan athrada i eryn “the man crosses the mountains [one way]”.
A prefix for “on both sides, across” in The Etymologies of the 1930s based the root ᴹ√AT(AT) “again, back” having to do with the number “two” (Ety/AT(AT)).
Conceptual Development: The root ᴹ√YAT “join” had deleted preposition N. iath “across” (EtyAC/YAT). In Tolkien’s later writings, prefixal “across” was generally thar or athra- based on the root √THAR (PE17/14), while S. ath- was used for “easy” (PE17/148).