Kent finished second in Division Two last season and were not given a spot in the top flight after Durham were relegated because of financial issues.
Gillespie, who joined Kent as cover for Allan Donald in March, led Yorkshire to promotion from Division Two in 2012.
"For this whole squad, the world is their oyster," he told BBC Radio Kent.
"Kent have shown that they can do it and they can challenge to play in the first division.
"They play against first division sides in white-ball cricket all the time so the lads know they can do it, it's just a couple of things falling into place."

A group of employees from the Northern Ireland Public Service Alliance (Nipsa) union are in dispute with their employer over pay and job grading.
The grievance involves 12 higher executive officers (HEOs) at Nispa offices in Belfast and Londonderry.
In a statement, Nipsa said it was "committed to resolving this dispute".
Nipsa employees are not permitted to become members of the union they work for, but the staff involved are members of another trade union, Unite.
Unite representative Kevin Kelly said it was a long-standing dispute and he accused Nipsa's governing general council of a "failure to practice a standard of industrial relations that they would expect of their officials".
"My members are disgusted that their employer, who routinely challenges poor industrial relations practice across Northern Ireland, is acting in a manner which no employer would dare," Mr Kelly said.
"This dispute is resolvable if the will of the employer existed.
"It is therefore with reluctance and sadness that our members have been forced to take the ultimate action left open to them, that is the withdrawal of their labour".
The affected Nipsa employees are to hold a one-day strike on Friday 30 October and will then begin a work-to-rule protest from Monday 2 November.
Nipsa general secretary Brian Campfield said: "Nipsa is aware of the concerns of this group of its staff and has been involved in negotiations with their trade union, Unite, about how the issues in contention can be resolved to the satisfaction of both parties.
"Nipsa has advised Unite that it is committed to resolving this dispute and is presently engaged in further work which will hopefully settle any outstanding matters."

The US Treasury said that Senat Shipping was providing "extensive support" to a North Korean company already under sanctions.
It means that any assets the firm holds in the US are frozen and prohibits US citizens from doing business with it.
Senat Shipping has denied the allegations.
The US Treasury makes clear that measures against Senat Shipping also extend to the company's president, Leonard Lai.
It said that Senat Shipping was co-operating with Ocean Maritime Management Company (OMMC), a North Korean firm already under sanctions.
In 2013, a ship operated by OMMC was seized by Panamanian authorities for hiding undeclared military equipment from Cuba under its cargo of sugar.
According to US authorities, Senat arranged the purchase, repair, certification, and crewing of vessels for OMMC.
"Arms shipments transported by OMMC serve as a key resource for North Korea's ongoing proliferation activities," said a US Treasury press release.
The United Nations and the US imposed sanctions on OMMC in July 2014 for the attempted arms import into North Korea.
Senat Shipping's statement confirmed that it had chartered OMMC vessels in the past to operate part of its business but says since then it had stopped dealing with all North Korean ship owners.
The firm denies all of the accusations brought against it by the US.
Company president Leonard Lai said the US move to put him and his company on the blacklist was "purely based on my historical dealings with OMMC and not based on anything illegal".
Under UN sanctions, North Korea is banned from weapons exports and the import of all but small arms. There are also separate US sanctions also target Pyongyang's nuclear arms programme.
Kent interim assistant coach Jason Gillespie says they can challenge for promotion to Division One of the County Championship this season.
[NEXT_CONCEPT]
Trade union staff are to take strike action against their own organisation, claiming it has been "acting in a manner which no employer would dare".
[NEXT_CONCEPT]
The US has blacklisted a Singapore-based shipping firm over allegations it is supporting illicit arms shipments to North Korea.