Mr Jones, whose standing with fellow Tory MPs since the Boris Johnson Partygate crisis first blew up was reflected on Monday in his election to the powerful 18-person executive of the 1922 committee, said he was looking for a candidate to "reinvigorate trust in politics" by fighting a fair campaign while respecting rival candidates.
That man, Mr Jones added, was former Chancellor of the Exchequer, Rishi Sunak.
"Standards and integrity must be central in public life whether you are a councillor, a Member of Parliament or a Prime Minister.
"I am by no means suggesting that one should not enter public life if you have received a parking ticket or are a multiple divorcee or maybe smoked pot at university. I am not seeking some sort of puritanical approach.
"The country needs someone who is consistent and transparent to reinvigorate trust in politics.
"The leader of any political party - in fact the leader of any organisation - needs certain qualities.
They should be able to listen to all views and, when they have done so, command with authority as they set a course of action and then follow through to deliver the policy.
"That position of command and respect is earned through honesty, application and ability.
"Rishi Sunak has told his supporters that he wants to promote his own candidacy in a positive way.
"He does not want to run down the other candidates.
"When it comes to it we are all colleagues and need to respect each other’s positions and row in behind whoever the new leader is allowing them to set their agenda.
"In backing Rishi I am aware that he received a fine for breach of Covid regulations.
"For most people that was a surprise as he turned up at Number 10 for a meeting and walked into the Cabinet room as the Prime Minister was being present with a cake. Once he received the fine he paid it.
There was no dissembling, no denial and no appeal even though many thought it was harsh.
"He accepted the ruling. I think people understand the difference between that and the many months of changing claims that preceded the Prime Minister’s fine."
There are currently ten candidates running to replace him after transport secretary Grant Shapps withdrew from the battle to support Rishi Sunak, but the 1922 Committee led by its chair Sir Graham Brady has set rules for the battle to replace Boris Johnson which should reduce that number to four or five by Friday.
Boris Johnson will remain as caretaker PM until September 5
As things stand Mr Sunak is in the lead for individual MP backing with at least 37 supporters.
Miss Penny Mordaunt is in second place and backbencher Tom Tugendhat has also passed the threshold, with the intentions of more than 150 Tory MPs confirmed up to now.
Andrew Jones MP, a long-time critic of Boris Johnson's behaviour over Partygate, was one of 148 Conservative MPs to vote against his PM in a vote of confidence in the House of Commons in early June.
In his view, the next leader needs to make tackling the cost of living crisis a priority but must also be someone who can command respect for Great Britain on the international stage.
Mr Jones said: "Because we face challenging times we need a Prime Minister who is strong and with proven economic ability and business credibility.
"The cost of living crisis has been brought about by a range of factors but the most noticeable impact in our everyday lives is inflation in everyday goods.
"Inflation is caused by an excess of demand over supply, and it was building up in the system.
"The economic aftermath of the pandemic was already causing big problems, but that has been tipped into a crisis by the invasion of Ukraine.
"We can see that with oil, gas and grain prices.
"So the new Prime Minister needs to look at those structural imbalances in our economy that make us susceptible to inflation.
" That means addressing our dependence on fossil fuels and boosting the productivity and capacity of the economy.
"While addressing these long-term structural problems he or she needs to protect those most affected by rapidly rising prices.
"This may be through the benefits system, through increasing the living wage, through education in how to lower personal and household costs or through a variety of other means.
"In the field of foreign affairs we need someone already known and respected on the international stage to keep the pressure on Russia to quit Ukraine.
As for the timing, some Tory MPs have called for the interim period in which Mr Johnson remains in power to be shortened.
But Mr Jones believes the timetable the Conservative Party has now adopted for the leadership election would allow for a proper debate without the campaigning dragging on.
Mr Jones said: The 1922 committee -the representative committee of Conservative backbenchers - has set a date for the end of the contest as September 5.
"That gives MPs the chance to choose two candidates to put to the Conservative Party membership over the summer and for those two people to lay out their stall to the public, too.
"I think that timetable recognises the need for a relatively quick succession without being hasty."
Indeed, the threshold set is quite quick and quite tough
MPs need 20 backers each by 6pm today, Tuesday, July 12 before they face a knockout vote by MPs tomorrow afternoon, Wednesday.
To even get to that point there is also a threshold of 30 votes to get through that first round vote.
The full list of candidates is: