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Nicole Scherzinger 's putting her home back on the market ... loosening up the button on the price -- bringing it down by half a million bucks!

The singer-songwriter relisted her Hollywood Hills home on Monday for $5,999,000 ... months after she took the house off the market to lease the place instead. Cris Nolasco

Nicole was asking for $6.5 million in May -- down from her $6.7 million asking price in March -- before renting the place for $35K a month ... not bad for some passive income!

The 4-story pad sits above the Sunset Strip ... with views spanning from Downtown Los Angeles all the way to the Pacific Ocean. Cris Nolasco

The 4-bedroom, 5-bathroom house comes with a sky-high lounge with a swing, a sleek lap pool,

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