The Conservative leadership candidate said a no-deal Brexit could change the situation on independence in Scotland but vowed to reject any request for a new vote on the issue.
He spoke out after a private meeting with Tory members at the Royal George Hotel in Perth ahead of hustings in the city with leadership rival Boris Johnson.
Mr Hunt said: "All of us have to do more but to date as I've been watching the no-deal preparations I would like to see more focus on that from Nicola Sturgeon.
• READ MORE: Jeremy Hunt: I won't grant second independence referendum even if SNP win Holyrood majority"(The Scottish Government) have been very disappointing in their preparations of a no-deal Brexit, which none of us want, but any responsible Government or authority in the United Kingdom should be taking seriously because there is that risk."
Questioned what more the Scottish Government should be doing, he said: "This isn't the time to go into details but I think they know, and we know, the areas where more can be done."
Questioned if a no-deal could change the situation on independence, the Foreign Secretary said: "If it's handled the wrong way, yes.
"But I'm going to make sure if we do end up with a no-deal Brexit that it's one where we leave no stone unturned in order to mitigate the impact for farmers, the fishing community, other businesses affected by Brexit whether north or south of the border."
Mr Hunt was also asked about 1,000 free milkshakes being handed out at Perth restaurants ahead of the hustings after Nigel Farage and Tommy Robinson were targeted with the drinks during the European election campaign.
Businessman Frank Burger-Seed said his restaurants would be handing out the free drinks and urged people to "drink responsibly"
Mr Hunt said: "Personally I quite like milkshakes with a burger and so do my kids but I think they should be consumed and not used as projectiles."