Crocodile is right, you certainly don't need anything like a 600W RMS amplifier to drive these speakers to high volumes. Connect a true 600W RMS amp to these speakers, max out the volume and you won't see the speakers for smoke.
Are these the speakers?
SPLBOX.3W 8Ohm Waterproof Mini Box Speaker Pair 300w RMS White
The specifications stated are misleading. Sadly, not all specs are true and some (like these, IMO) are intended to mislead. For instance...
"
Speaker Maximum Recommended Power RMS (Watts) (One Speaker, Using Appropriate Crossover) = 150"
Appropriate crossover? What's that all about? Is the suggestion that these speakers need to be fitted with a high pass filter (to remove bass) to meet the 150W RMS power handling spec? Actually, that's likely to be true, IMO.
And...
"System Frequency Response (Hz) (Indicative Of Usable Range But Not Suitability) = 40-20000"
And...
"Full Speaker System Recommended RMS (Watts) (Using Pair/Full Set Of Speakers Not Necessarily Included & Appropriate Crossover) = 300"
Absolute tosh, IMO of course.
Oh, and did you read between the
lies (Oops, Freudian slip. I meant to say 'lines') of the blurb...
"Please note that the speaker has a white painted metal grille which will need to be regularly maintainted to ensure long term good looks."
i.e. the grills will rust if they so much as sniff moisture, IMO.
If the quoted 95dB/W @ 1m sensitivity and 8 Ohm impedance figures are correct, then your mini-amp should drive them to high volumes but as you've found, it doesn't so the stated specs are garbage (or maybe your mini amp is?).
My low powered (about 8 W RMS/channel) class T mini-amp drives small inefficient speakers to moderate volumes before clipping but it's not a particularly pleasant sound as I guess it's struggling at anything above low volumes.
Any half decent 30W to 50W RMS/channel into 8 Ohm amp (and I mean a real 30 - 50W amp - not one with fake specs) should be fine for this type of speaker, assuming the speakers are working correctly. If a 30W amp doesn't give the volume you desire then I suspect you've chosen the wrong speaker for your application - or the speakers are duds.