UPDATE 2-Thai PM promises more vaccine amid criticism of slow rollout

* New doses in addition to about 65 million ordered

* Vaccines push comes as bigger infection wave hits

* Thailand trailing many countries in vaccinations

* Private sector welcomes new vaccine procurement (Adds private sector response, comment from expert on Thai vaccine strategy)

BANGKOK, April 21 (Reuters) - Thailand is trying to secure 35 million more doses of COVID-19 vaccines from two or three companies this year on top of existing orders of around 65 million doses, Prime Minister Prayuth Chan-ocha said on Wednesday.

The government push comes amid growing public frustration about the slow vaccine rollout, with only 604,947 people inoculated so far, less than 1% of its population.

Of the new shots sought, the private sector will help source 10 to 15 million doses, Prayuth said in a Facebook post.

"I have ordered that we distribute and administer all the vaccines that we can find by December," he added.

Prayuth did not name the brands, nor specify whether the 35 million included the five to 10 million doses of Pfizer and BioNTech's vaccine that he announced were being sought on Tuesday.

After early success in containing COVID-19, Thailand is fighting a new wave that includes the highly transmissible B.1.1.7 variant.

The new outbreak has accounted for more than a third of Thailand's 46,643 cases, of which 110 were fatalities. It reported 1,458 new infections and two deaths on Wednesday.

Its mass vaccination plan has been centred on 61 million doses of locally-made AstraZeneca vaccine, the first batches scheduled for June.

For initial inoculations, Thailand has received 2 million doses of Sinovac Biotech's vaccine and has ordered a further 1.5 million doses, with 500,000 to arrive on Saturday and the rest next month. It has also imported 117,000 doses of AstraZeneca's vaccine.

The private sector, which had been for months seeking permission to import vaccines, welcomed its inclusion.

"This is the race. We have to be fast and have volume, but we haven't had any other vaccines," Chalerm Harnphanich, president of the Private Hospital Association, told Reuters.

Health experts also questioned the government's decision not to use the COVAX international vaccine-sharing facility, which has provided vaccines to 100 countries including the Philippines, Vietnam and South Korea.

"We missed an opportunity by not joining COVAX," said Thiravat Hemachudha, head of the Thai Red Cross Emerging Infectious Diseases Centre.

The government has argued joining COVAX risked higher costs and uncertain delivery times. (Reporting by Panu Wongcha-um and Panarat Thepgumpanat, Chayut Setboonsarng and Patpicha Tanakasempipat; Editing by Martin Petty)