-The IO should have issued you I-94 duration to the date of your passport expires (明年六月).
Two choices:
1. Go to the nearest CBP to see whether they can change your i-94 duration to 明年六月, and then get the new passport and extend the h1 before 明年六月.
2. If Choice i does not work, file an H1 extention petition before your current I-94 expires (本月底) with your current passport. The short effective term of the passport does not affect your H1 extention within U.S.
必须拿着新护照出国再进来?
-Not really. See above answer.
还是律师在美国国内可以搞定?
-Your employer can extedn your H1 within U.S.