The Australian carded a seven-under-par 65 in the second round to take his overall standing to 11-under.
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Rumford's only blemish was a bogey on the 12th hole, but his eight birdies moved him four places up the field to join overnight leader Uihlein, who finished four-under for the day following six birdies and two bogeys.
Louis Oosthuizen had an impressive round as he shot eight birdies, four of which were consecutive from the 12th hole, and moved one shot behind the joint leaders.
Rhein Gibson remains two shots off the pace following a second-round 68.