It's a little on the cumbersome side, but I believe this should work (without the extra join). This assumes that you can choose a character that will never appear in the field in question, to act as a separator.
You can do it without nesting the select, but I find this a little cleaner that having four references to SYS_CONNECT_BY_PATH.
select id,
parent_id,
case
when lvl <> 1
then substr(name_path,
instr(name_path,'|',1,lvl-1)+1,
instr(name_path,'|',1,lvl)
-instr(name_path,'|',1,lvl-1)-1)
end as name
from (
SELECT id, parent_id, sys_connect_by_path(name,'|') as name_path, level as lvl
FROM tbl
START WITH id = 1
CONNECT BY PRIOR id = parent_id)