I've been offered a job with a starting salary lower than what I was expecting, so I countered their offer with a starting salary that was about 10k higher. They ended up ~~meeting me half way~~ countering my counteroffer with a new offer that was in the middle, but I am not fully satisfied and want a bit more.

How do I go back asking them for to just bump it up a bit? I am willing to take a signing bonus as opposed to the additional salary increase. How do I go about going from their counter offer to asking for a signing bonus?