The off-spinner took four for 61 in 21.4 unchanged overs yesterday, extending his match haul to eight wickets, before bad light ended the fourth day with New Zealand 154 for six.
That left the Black Caps still needing a further 310 runs to reach their victory target of 468.
The most any side has ever made in the fourth innings to win a Test is West Indies' 418 for seven against Australia in Antigua in 2002/03.
And the corresponding record for New Zealand is the 325 for four they posted against Pakistan in Christchurch in 1993/94.
In that case they might rue both captain Alastair Cook's decision not to enforce the follow-on on Sunday and indeed bat on after lunch yesterday in a match where the whole of Friday's first day was washed out without a ball bowled.
For New Zealand, only former captain Ross Taylor, with a stylish 70, offered much in the way of significant resistance yesterday before he was yorked by Swann to leave New Zealand 154 for six.
In their first innings, New Zealand collapsed to 174 all out -- although this was an improvement on the 68 they managed in the second innings of their 170-run first Test defeat at Lord's.
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