What you're talking about here is 'macro' photography, which normally refers to photos at or around 1:1 image/subject ratio (ie the size of the image projected onto the sensor is "life sized".
Your Sigma gets pretty close (1:2 I think), but to get really close you need things like extension tubes, reversing rings, or a dedicated macro lens. I'll be interested to hear what the other closeup addicts reckon, but personally, having mucked around with closeup filters and macro converters years ago, having a proper macro lens is pure joy. Lenses like the £250 Tamron 90mm 2.8, Sigma 105mm 2.8 , or a second hand Nikon 105mm 2.8 are pretty much the sharpest lenses money can buy.
As for tripods...hmmmmmmm
For macro work, you need something very flexible, so either tripod with a reversible centre column, or a 'bent bolt' type that can be splayed out in wierd ways so you can get low to the ground. Anyone got any suggestions, as I daren't suggest my setup since he said the words "on a budget"

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The best I can come up with is a Benbo mini trekker at £60 with a £30 Manfrotto ball head