He has issued a come and get me plea on social media.
New Hawks manager Paul Doswell made it known that he intends to introduce three extra morning training sessions a week next season.
Carter who recently started a new career in recruitment is unable to make that commitment.
The highly respected and popular 27-year-old wants to stay local and is sure to attract a lot of interest from clubs.
He said: ‘I made it known beforehand to the Hawks that I can’t consider going full-time.
‘Starting a new career in recruitment means it is not the right time to do that.
‘It would make no sense for me and I have to consider my long-term prospects.
‘I never really heard much until I saw the press conference with the manager on social media.
‘Then Shaun Gale was given the job of ringing around some of the players telling them there was nothing at the club for them.
‘I am going to leave my phone on for a couple of weeks and see what happens.
‘It has been busy already and I have spoken with a few people.
‘I will wait and weigh up my options before making a decision.
‘Things like the level of football and the amount of travelling will have to be taken into account.’
Carter joined the Hawks in January 2017 following a two-and-a-half-year spell at Gosport Borough.
Previous to that in 2014 he helped Sholing lift the FA Vase at Wembley.
Carter spent a month back at Gosport on loan at the start of the year after knee surgery and five months on the sidelines.
He showed in the final stages of the season that he has fully recovered.
His tenacious tackling, hard work and ability to read the game will prove an asset wherever he ends up.
He provides a combative presence in the middle of the park.
Local clubs will also be keen to contact other Hawks players who have been released or unable to commit to full-time.
These include Rory Williams, Jordan Rose and Ryan Woodford.
All three are attractive propositions with their National League experience.