Semiprime numbers which, along with their prime factors, generate many semiprimes by concatenation

You missed a nice one, $9$, which also happens to be what you call a "super STB number". The next one is indeed $1191$. With just a little more of your impressive patience, you could have got up to $3497$, which is the next one. After that there are none up to $10000$. Here's the code.

P.S.: Since you wanted a square-free "super STB number", I took out the prime precalculation so it can run up to larger numbers -- it doesn't find any further "super STB numbers", but here are the next few "STB numbers":

• $9$, $1191$, $3497$, $28267$, $50191$, $60693$, $65049$, $92823$, $98759$, $212523$, $241419$, $243611$, $256693$, $281949$, $292683$, $324699$, $368587$, $383831$, $403891$, $460783$, $497923$, $538413$, $560523$, $572569$, $670733$, $798061$, $850233$, $858597$, $878079$, $904079$, $984909$, $1091823$, $1097371$, $1128381$, $1160889$, $1201631$, $1337861$, $1352527$, $1436857$, $1492233$, $1554421$, $1605007$, $1724303$, $1787353$, $1796917$, $1904907$, $1980571$, $2002393$, $2502017$, $2508981$, $2533809$.

It's nice that one of them, $1352527$, is quite similar to the number in the URL for this question, $1357327$ :-) Here's the new code.

P.S.: I accidentally left the program running, so I might as well give you the output :-)

• $9^*$, $1191$, $3497$, $28267$, $50191$, $60693$, $65049$, $92823$, $98759$, $212523$, $241419$, $243611$, $256693$, $281949$, $292683$, $324699$, $368587$, $383831$, $403891$, $460783$, $497923$, $538413$, $560523$, $572569$, $670733$, $798061$, $850233$, $858597$, $878079$, $904079$, $984909$, $1091823$, $1097371$, $1128381$, $1160889$, $1201631$, $1337861$, $1352527$, $1436857$, $1492233$, $1554421$, $1605007$, $1724303$, $1787353$, $1796917$, $1904907$, $1980571$, $2002393$, $2502017$, $2508981$, $2533809$, $2631211$, $2676763$, $3231581$, $3295259$, $3415701$, $3460633$, $3511867$, $3534319$, $3537017$, $3544993$, $3606951$, $3704257$, $3743511$, $4034281$, $4223281$, $4338599$, $4471643$, $4490169$, $4900039$, $5041083$, $5143289$, $5278933$, $5361301$, $5457649$, $5488633$, $5537143$, $5580537$, $6194077$, $6245753$, $6317041$, $6352233$, $6386857$, $6416587$, $6518283$, $6544743$, $6876867$, $6916179$, $6959059$, $6963317$, $7017181$, $7099041$, $7099499$, $7111821$, $7188393$, $7237221$, $7419589$, $7445963$, $7473563$, $7520519$, $7646719$, $7774019$, $7775133$, $7801449$, $7842377$, $7847481$, $8344171$, $8472817$, $8519701$, $8629539$, $8649169$, $8778453$, $8880303$, $8904191$, $9124159$, $9235793$, $9304651$, $9413697$, $9470443$, $9596253$, $9601181$, $9683533$, $10013413$, $10122031$, $10218503$, $10228737$, $10275819$, $10508923$, $10546027$, $10571393$, $10706149$, $10983659$, $11135127$, $11409817$, $11413603$, $11840187$, $11984761$, $12250457$, $12291113$, $12707009$, $12864441$, $12869607$, $12887569$, $13285843$, $13427363$, $13593939$, $13612357$, $13616731$, $13749113$, $13840457$, $13856361$, $13929567$, $13971561$, $14068167$, $14160849$, $14415355$, $14541711$, $14665947$, $14675741$, $14682667$, $14721403$, $15237173$, $15247189$, $15278659$, $15978491$, $16068153$, $16188623$, $16253429$, $16475807$, $16817781$, $17137013$, $17200849$, $17220387$, $17257609$, $17269721$, $17355643$, $17472831$, $17521697$, $18325497$, $18454411$, $18717729$, $18987211$, $19021153$, $19197173$, $19346707$, $19369257$, $19457853$, $19649451$, $19930767$, $20232971$, $20237883$, $20549301$, $20783589$, $20795353$, $21245551$, $21288091$, $21342259$, $21370831$, $21377323$, $21378759$, $21508671$, $21525769$, $21692599$, $21694081$, $21718173$, $21777757$, $21785019$, $21858701$, $21933339$, $22366471$, $22508801$, $23102593$, $23223363$, $23256039$, $23261671$, $23535079$, $23588601$, $23660751$, $23726167$, $23741449$, $23863611$, $24043623$, $25212583$, $25227681$, $25956257$, $26055613$, $26093989$, $26564001$, $26713171$, $26824867$, $27000019$, $27219883$, $27639139$, $27722639$, $27929599$, $28087531$, $28222149$, $28419969$, $28639041$, $28935393$, $29055037$, $29678497$, $29721687$, $29738893$, $30287487$, $30702939$, $30945919$, $31125587$, $31376841$, $31495791$, $31503481$, $31513211$, $31620629$, $31767303$, $32120399$, $32141469$, $32266537$, $32281377$, $32545351$, $32905833$, $32970279$, $32999519$, $33412559$, $33550297$, $34020253$, $34262203$, $34773663$, $34890133$, $35436679$, $35493779$, $35536589$, $35683259$, $36177457$, $36376707$, $36449013$, $36548923$, $36607723$, $36742477$, $36825589$, $37072597$, $37420797$, $37677859$, $37800393$, $38008799$, $38030773$, $38319139$, $38400697$, $38552677$, $38689303$, $38722053$, $39021271$, $39048419$, $39291227$, $39296051$, $39558039$, $39678463$, $39685453$, $39870669$, $40036901$, $40508027$, $40578033$, $40606631$, $41187439$, $41212153$, $41733863$, $41877861$, $42094451$, $42139753$, $42178573$, $42401609$, $43090741$, $43237633$, $43337713$, $43474017$, $44476527$, $44922211$, $44973391$, $45307347$, $45721821$, $45812239$, $45849463$, $46510131$, $46676083$, $46784181$, $46816351$, $46825437$, $47199921$, $47440119$, $48135067$, $48136157$, $48877183$, $49291493$, $49293211$, $49460729$, $49580227$, $49782961$, $49912117$, $50373701$, $50476351$, $50902987$, $50988633$, $51400743$, $51523909$, $51873643$, $52026787$, $52337713$, $52517931$, $52835011$, $52873773$, $52874979$, $52962817$, $52997281$, $53999097$, $54370977$, $54635397$, $55059789$, $55735993$, $55980659$, $56234401$, $56403967$, $56436697$, $56533543$, $56853373$, $56979277$, $57012043$, $57065371$, $57166367$, $57338329$, $57667693$, $57990099$.

The asterisk is for a "super STB number", so unfortunately no more of those, but plenty of "STB numbers".