That, I should have done. I thought to save the number of lines previously read so that I could skip them; I don't know why it didn't occur to me to just save the whole damn offset.

I used this trick the other day -- compute byte range partitions on a large file and use fseek when processing each. Is there a standard unix program that can output an arbitrary byte range from a seekable input? At first blush dd(1) looked like the ticket, but block-oriented operation means extra invocations to deal with a byte range that's not necessarily block-aligned.

tail -c offset [input] | head -c length

will do the job.

You can set dd's "bs" parameter to 1, then seek, skip & count are all in terms of bytes.