Promoted Billericay went down 2-0 at Hardenhuish Park to a Chippenham Town side who have now won three and lost three.
Second half goals from Nat Jarvis and James Guthrie after 54 and 65 minutes respectively won it for the Bluebirds.
Only Concord Rangers and Eastbourne Borough remain unbeaten as both drew on Saturday.
The Beachboys shared a 1-1 draw at Grosvenor Vale against Wealdstone, whose Danny Green opened the scoring after 20 minutes and in-form Alex Wall scored again for Rangers 15 minutes later.
It was a similar tale at Clarence Park where the goals came early – Ralston Gabriel for the St Albans City after 11 minutes and Michael West 3 minutes later for Boro.
At the other end of the table, Weston-super-Mare and Truro City are still searching for their first wins.
Weston picked up a point from a 1-1 Woodspring Stadium draw with Oxfod City, whose Kabongo Tshimanga cancelled out Jarrad Welch`s 72nd minute opener for the Seagulls in the 83rd minute.
Truro lost 3-1 at leaders Woking, despite taking an early 6th minute lead through Jamie Richards.
Kane Ferdinand eventually equalised after 64 minutes and 3 minutes later, Jack Cook put the hosts in front, with Max Kretzschmar making sure with a Cards third in the 77th minute.
Relegated Torquay United were beaten 2-0 at Plainmoor by Hampton & Richmond Borough, for whom Rhys Murrell-Williamson after 25 minutes and then Michael Corcoran 7 minutes from time were on the mark.
And East Thurrock United are up and running with their first win and points, thanks to a 79th minute goal from Danilo Orsi-Dadomo at home to promoted Slough Town.
Hungerford Town were another side to record their opening win of 2018/19, 1-0 at Bulpit Lane against Dartford.
Connor Lynch`s goal just shy of the hour-mark proved enough for the Crusaders.
Chelmsford City won 1-0 at Melbourne Community Stadium against ten-man Bath City.
The visitors had Robbie Cundy dismissed a minute before half-time for denying a clear goalscoring opportunity and Rhys Murphy despatched the resulting penalty.
Despite their extra man, Chelmsford were unable to add to their tally in the second period.
Jordan Parkes gave Hemel Hempstead Town an early 8th minute lead at Park View Road against Welling United.
But 9 minutes later, Brendan Kiernan replied which was eventually enough to earn a point.
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