Dr Smith resigned from cabinet after the Herald revealed he had, as ACC Minister, written a reference for ACC claimant Bronwyn Pullar who unsuccessfully sought payment from ACC.
It is possible he could be promoted back to the ministry in a future reshuffle. The next one is expected this year when Speaker Lockwood Smith is named as High Commissioner to London.
Mr Carter is one of the prime contenders for the Speakership and so he could hand Local Government back to Dr Smith at that time.
Mr Bridges' elevation leaves a vacancy as chairman of the finance and expenditure committee, which is usually considered a stepping stone for promotion.
National list MP Paul Goldsmith is the current deputy although has been in Parliament only five months.
Prime Minister John Key announced the reshuffle this afternoon.
Mr Bridges, who is in his second term of Parliament, will be sworn in tomorrow.