Senior politicians who served under Ed Miliband would be brought back to the Labour frontbench should Jess Phillips win the party’s leadership election.

The backbench MP has said she would reappoint the former shadow home secretary Yvette Cooper and former shadow work and pensions secretary Rachel Reeves, who both left their positions under Jeremy Corbyn, to her shadow cabinet.

Phillips said the pair, who now chair select committees, had enormous intellect and experience and would be an asset to her leadership team. The former chair of the women’s parliamentary Labour party is pledging to create a shadow cabinet that is “at least 50% women” if she wins.

She made the comments at an event at the Lowry in Salford with the broadcaster Iain Dale after it was confirmed she has secured the 22 nominations from MPs to go on to the next stage of the Labour leadership contest.

Lisa Nandy, MP for Wigan, Keir Starmer, the shadow Brexit secretary, and Rebecca Long-Bailey, the shadow business secretary, also made it through to the first round of the election process, with nominations closing at 2.30pm on Monday.

Louise Haigh, the shadow policing minister, would also be offered a role, Phillips told the audience. “Any time Lou stands up in the House of Commons to speak, I listen, because she is always right,” she said.

Long-Bailey could also be offered a job on Phillips’s frontbench. The leadership rivals are perceived to be the furthest apart ideologically, with Phillips a well-known Corbyn critic who has won nominations from those considered to be on the right of the party, while Long-Bailey, a socialist, has been described as the Corbyn “continuity candidate”.

Those with 22 nominations must now try and seek the support of either 5% of all constituency Labour parties or three Labour affiliates – of which at least two must be trade unions.

A YouGov poll in late December among members placed Phillips in third behind Starmer and Long-Bailey.

Part of Phillips’s pitch as leader is a forward-thinking childcare policy. She said she would set up a national childcare service offering universal, Scandinavian-style care.

This would be for three- and four-year-olds, then extended to two-year-olds. She said it would replace a series of “baffling rebates, vouchers, tax incentives and entitlements that vary from town to town”. It would not distinguish between working and non-working parents.

Currently, two-year-olds from disadvantaged families get 15 hours of free childcare but as soon as their parents go back to work they are no longer eligible. For those in work with three- and four-year-olds they can access 30 hours of free childcare.

Like Corbyn, Phillips said she would want to revive Sure Start services after the closure of 1,000 centres since 2010. Labour’s 2019 manifesto pledged to support families by providing two- to four-year-olds with 30 hours of free pre-school education per week, eventually incorporating one-year-olds, with additional hours at subsidised rates.

Phillips said: “Nothing is more important than making sure our kids have a better life than we did. But we’re going backwards and people are losing hope. After the war, the 1945 Labour government created the NHS. It was a simple idea that none of us can imagine living without: nobody should be ill because they can’t afford medical care.

“In the years ahead I want people to look back and ask: how did we ever manage before there was national universal childcare?”