The American English /r/ phoneme is rounded, so there should be some "W" sound.

The problem is likely insufficient retroflexion. Make sure your tonguetip is curled up and back, so that you can touch your hard palate with the bottom of the tonguetip. With your tongue in this position and your lips rounded, say [ə]. In practice, native English speakers don't actually touch anything with their tongue when saying /r/, but the tongue is almost there and that's the position, so you can practice it.

For further phonetics, let me recommend J. C. Catford's helpful little book A Practical Introduction to Phonetics; it's designed for the autodidact and is full of little phonetic experiments you can do.

Rule: Use a Dictionary

Yes, there is a rule, and that rule is that you must look them up in a dictionary if you are not a native speaker.

That’s because words beginning with re- in English can, depending on the word, be pronounced with any of eight different vowels:

1. /ra/
2. /rɑ̃/
3. /rɒ/
4. /re/
   
   
5. /rə/
6. /rɛ/
7. /ri/
8. /rɪ/

The last three or four at the end of that list tend to be for native words, while the ones at the beginning tend to be for unassimilated imports.

But in diphthongs like reindeer or reynard — let alone reiter or rearward — all bets are off.

Examples

Here’s an alphabetized sample list, with pronunciation following:

• readable (adj.) /ˈriːdəb(ə)l/
• ready (v.) /ˈrɛdɪ/
• rearward (adv.) /ˈrɪɚwɚd/
• reasonable (adj.) /ˈriːz(ə)nəb(ə)l/
• rebel (adj.) /ˈrɛbəl/
• rebel (v.) /rɪˈbɛl/
• rebuff (v.) /riːˈbʌf/
• rebuff (v.) /rɪˈbʌf/
• recapture (v.) /riːˈkæptjʊə(r)/
• recherché (adj.) /rəʃɛrʃe/
• recollet (v.) /rekɔle/
• recueillement (n.) /rəkœjmɑ̃/
• redact (v.) /rɪˈdækt/
• redolence (n.) /ˈrɛdələns/
• redondilla (n.) /redonˈdiʎa/
• redress (v.¹) /rɪˈdrɛs/
• redress (v.²) /riːˈdrɛs/
• refectory (n.) /rɪˈfɛktərɪ/
• refect (v.) /rɪˈfɛkt/
• regime (n.) /reɪˈʒiːm/
• regiment (n.) /ˈrɛdʒɪmənt/
• reign (n.) /reɪn/
• reindeer (n.) /ˈreɪndɪɚ/
• reis (n. pl.) /reɪs/
• reis (n.) /raɪs/
• reiter (n.) /ˈraɪtɚ/
• relevé (n.) /rələve/
• remake (n.) /ˈriːmeɪk/
• remake (v.) /riːˈmeɪk/
• remarque (n.) /rəmark/
• remboîtage (n.) /rɑ̃bwataʒ/
• remise (n.) /rəmiz/
• remoulade (n.) /remulad/
• remplaçant (n.) /rɑ̃plasɑ̃/
• rendezvous (n.) /ˈrɒndɪvuː/, /ˈrandəvu/, /rɑ̃devu/
• res (gen.) /reɪz/
• resolve (n.) /rɪˈzɒlv/
• resounding (ppl. a.) /rɪˈzaʊndɪŋ/
• resurrect (v.) /rɛzəˈrɛkt/
• retrieval (v.) /rɪˈtriːvəl/
• reunion (n.) /riːˈjuːnɪən/
• revolutionize (v.) /rɛvəˈl(j)uːʃənaɪz/
• revisit (v.) /riːˈvɪzɪt/
• reynard (n.) /ˈreɪnɚd/
• rez-de-chaussée (n.) /redʃose/

And here grouped by pronunciation:

/ra/

• reis (n.) /raɪs/
• reiter (n.) /ˈraɪtɚ/
• rendezvous (n.) /ˈrɒndɪvuː/, /ˈrandəvu/, /rɑ̃devu/

/rɑ̃/

• remboîtage (n.) /rɑ̃bwataʒ/
• remplaçant (n.) /rɑ̃plasɑ̃/
• rendezvous (n.) /ˈrɒndɪvuː/, /ˈrandəvu/, /rɑ̃devu/

/rɒ/

• rendezvous (n.) /ˈrɒndɪvuː/, /ˈrandəvu/, /rɑ̃devu/

/re/

• recollet (v.) /rekɔle/
• redondilla (n.) /redonˈdiʎa/
• regime (n.) /reɪˈʒiːm/
• reign (n.) /reɪn/
• reindeer (n.) /ˈreɪndɪɚ/
• reis (n. pl.) /reɪs/
• remoulade (n.) /remulad/
• res (gen.) /reɪz/
• reynard (n.) /ˈreɪnɚd/
• rez-de-chaussée (n.) /redʃose/

/rə/

• recherché (adj.) /rəʃɛrʃe/
• recueillement (n.) /rəkœjmɑ̃/
• relevé (n.) /rələve/
• remarque (n.) /rəmark/
• remise (n.) /rəmiz/

/rɛ/

• ready (v.) /ˈrɛdɪ/
• rebel (adj.) /ˈrɛbəl/
• redolence (n.) /ˈrɛdələns/
• regiment (n.) /ˈrɛdʒɪmənt/
• resurrect (v.) /rɛzəˈrɛkt/
• revolutionize (v.) /rɛvəˈl(j)uːʃənaɪz/

/ri/

• readable (adj.) /ˈriːdəb(ə)l/
• reasonable (adj.) /ˈriːz(ə)nəb(ə)l/
• rebuff (v.) /riːˈbʌf/
• recapture (v.) /riːˈkæptjʊɚ/
• redress (v.²) /riːˈdrɛs/
• remake (n.) /ˈriːmeɪk/
• remake (v.) /riːˈmeɪk/
• reunion (n.) /riːˈjuːnɪən/
• revisit (v.) /riːˈvɪzɪt/

/rɪ/

• rearward (adv.) /ˈrɪɚwɚd/
• rebel (v.) /rɪˈbɛl/
• rebuff (v.) /rɪˈbʌf/
• redact (v.) /rɪˈdækt/
• redress (v.¹) /rɪˈdrɛs/
• refect (v.) /rɪˈfɛkt/
• refectory (n.) /rɪˈfɛktərɪ/
• resolve (n.) /rɪˈzɒlv/
• resounding (ppl. a.) /rɪˈzaʊndɪŋ/
• retrieval (v.) /rɪˈtriːvəl/