Upgrade PSU for 1070?

Solution
There's no need.

Even with a very power hungry CPU you wouldn't have issues. The GTX1070 likely won't hit 200W usage, and the rest of the system is likely under 200W.

Give the CPU if you want, and whether it's overclocked or not (the rest doesn't matter too much) but it's not really an issue.

Update:
That TIER list doesn't seem to conform much to reality in terms of power supply failure. The CX600M seems to do quite well with customer feedback, and reviews vary though you get ones like THIS:
http://www.guru3d.com/articles_pages/corsair_cx600m_psu_review,8.html

"Again, stability wise we have very little to complain about..."

and...

apk24

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A CX600M is rated to deliver enough power for the GTX 1070 and you will probably be just fine. However, it is generally not recommended for gaming/workstation builds.As long as it doesn't overheat and you don't try to draw more than 550-600W it'll last till you're forced to upgrade.

For your future reference on the quality of PSUs: PSU tier list. The CX600M is a tier 4 PSU.
For system power requirements, check out
table:POWER REQUIREMENTS FOR GRAPHICS CARDS
calculator:OuterVision Power Supply Calculator


tl;dr: Upgrade if you have problems, keep using it until you do.
 
There's no need.

Even with a very power hungry CPU you wouldn't have issues. The GTX1070 likely won't hit 200W usage, and the rest of the system is likely under 200W.

Give the CPU if you want, and whether it's overclocked or not (the rest doesn't matter too much) but it's not really an issue.

Update:
That TIER list doesn't seem to conform much to reality in terms of power supply failure. The CX600M seems to do quite well with customer feedback, and reviews vary though you get ones like THIS:
http://www.guru3d.com/articles_pages/corsair_cx600m_psu_review,8.html

"Again, stability wise we have very little to complain about..."

and
http://www.kitguru.net/components/power-supplies/zardon/corsair-builder-series-modular-cx600m-psu-review/7/

"The unit didn’t exhibit any glaring technical faults which could cause a reservation – there are actually no concerns and performance throughout all our testing was certainly commendable."

So... what's the big deal?

From what I can tell one of the main issues is delivering stable power nearer to 100% load which you won't be doing anyway with a single GTX1070.
 
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