The recently-born boy was found near Shore Road, Hythe, on Thursday.
Police have now arrested a 36-year-old woman on suspicion of murder.
She is currently in police custody.
Detective Chief Inspector Liz Williams said: ‘This is a difficult and complex investigation, and we have a dedicated team working hard to establish the exact circumstances of what took place.
‘Our officers are carrying out house-to-house enquiries in the area so if you do have any concerns, then please speak to them.
‘If you do have information that you think could help our enquiries, then please come forward and speak to us.’
Anyone with information can call us on 101, quoting Operation Verse, or the reference number 44200084127.
You can also submit a report using the police’s online portal here.
Alternatively, you can call Crimestoppers anonymously on 0800 555 111.