We have raised a similar question recently with the support team. There doesn't seem to be a way to adjust that automatically. At the moment, we change the shipping price manually on the channel, and then use the rule engine to convert the shipping to Saturday service, for the specific days, using received hour amongst other things as a condition.
The rule also has to be turned on and off every week. It is somewhat inconvenient.
Is there anyway to get Linnworks to change the shipping template after 3pm on a Thursday to increase the cost of one day premium shipping?
We use DPD for next day and they charge a premium for Saturday delivery that if we don't increase our costs can mean we lose money on sales, at the moment its a manual process to go in and update the template every week